Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 64.5%

Time: 29.6s

Alternatives: 20

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(t + \frac{-2 - \frac{n \cdot \left(U - U*\right)}{Om}}{\frac{Om}{l\_m \cdot l\_m}}\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m}{\frac{Om}{l\_m}}\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(-2 \cdot \frac{U}{Om} + \frac{U \cdot \left(n \cdot \left(U* - U\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* U (* 2.0 n)))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_3 0.0)
     (*
      (sqrt
       (*
        (* 2.0 n)
        (+ t (/ (- -2.0 (/ (* n (- U U*)) Om)) (/ Om (* l_m l_m))))))
      (sqrt U))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (/ l_m (/ Om l_m)))) t_1)))
       (*
        (* l_m (sqrt 2.0))
        (sqrt
         (* n (+ (* -2.0 (/ U Om)) (/ (* U (* n (- U* U))) (* Om Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (t + ((-2.0 - ((n * (U - U_42_)) / Om)) / (Om / (l_m * l_m)))))) * sqrt(U);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m / (Om / l_m)))) + t_1)));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (t + ((-2.0 - ((n * (U - U_42_)) / Om)) / (Om / (l_m * l_m)))))) * Math.sqrt(U);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m / (Om / l_m)))) + t_1)));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = U * (2.0 * n)
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (t + ((-2.0 - ((n * (U - U_42_)) / Om)) / (Om / (l_m * l_m)))))) * math.sqrt(U)
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m / (Om / l_m)))) + t_1)))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(-2.0 - N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(N[(-2.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(U * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $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 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 41.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

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

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

   6. Applied egg-rr 55.3%

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

   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 70.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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 75.1%

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

   4. Applied egg-rr 75.1%

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

   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 33.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 38.2%

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

   6. Applied egg-rr 38.2%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

   9. Simplified 11.9%

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

   10. Taylor expanded in l around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 27.2%

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

   12. Simplified 27.2%

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

4. Final simplification 64.1%

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

Alternative 2: 62.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.5 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{l\_m}{Om} \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;l\_m \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.02 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\left(\frac{l\_m}{Om} \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)\right) \cdot \left(U \cdot l\_m\right) + U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(-2 \cdot \frac{U}{Om} + \frac{U \cdot \left(n \cdot \left(U* - U\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.5e-184)
   (sqrt
    (* (+ t (* (* (/ l_m Om) (* n (/ l_m Om))) (- U* U))) (* n (* 2.0 U))))
   (if (<= l_m 3.4e+33)
     (sqrt
      (* (* 2.0 U) (* n (+ t (/ (* (* l_m l_m) (- (* U* (/ n Om)) 2.0)) Om)))))
     (if (<= l_m 1.02e+239)
       (sqrt
        (*
         n
         (*
          2.0
          (+
           (* (* (/ l_m Om) (+ -2.0 (/ n (/ Om (- U* U))))) (* U l_m))
           (* U t)))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt
         (* n (+ (* -2.0 (/ U Om)) (/ (* U (* n (- U* U))) (* Om Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.5e-184) {
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else if (l_m <= 3.4e+33) {
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	} else if (l_m <= 1.02e+239) {
		tmp = sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.5d-184) then
        tmp = sqrt(((t + (((l_m / om) * (n * (l_m / om))) * (u_42 - u))) * (n * (2.0d0 * u))))
    else if (l_m <= 3.4d+33) then
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((l_m * l_m) * ((u_42 * (n / om)) - 2.0d0)) / om)))))
    else if (l_m <= 1.02d+239) then
        tmp = sqrt((n * (2.0d0 * ((((l_m / om) * ((-2.0d0) + (n / (om / (u_42 - u))))) * (u * l_m)) + (u * t)))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt((n * (((-2.0d0) * (u / om)) + ((u * (n * (u_42 - u))) / (om * om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.5e-184) {
		tmp = Math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else if (l_m <= 3.4e+33) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	} else if (l_m <= 1.02e+239) {
		tmp = Math.sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.5e-184:
		tmp = math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))))
	elif l_m <= 3.4e+33:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))))
	elif l_m <= 1.02e+239:
		tmp = math.sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.5e-184)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(l_m / Om) * Float64(n * Float64(l_m / Om))) * Float64(U_42_ - U))) * Float64(n * Float64(2.0 * U))));
	elseif (l_m <= 3.4e+33)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(Float64(l_m * l_m) * Float64(Float64(U_42_ * Float64(n / Om)) - 2.0)) / Om)))));
	elseif (l_m <= 1.02e+239)
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(Float64(Float64(Float64(l_m / Om) * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))) * Float64(U * l_m)) + Float64(U * t)))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(n * Float64(Float64(-2.0 * Float64(U / Om)) + Float64(Float64(U * Float64(n * Float64(U_42_ - U))) / Float64(Om * Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.5e-184)
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	elseif (l_m <= 3.4e+33)
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	elseif (l_m <= 1.02e+239)
		tmp = sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U / Om)) + ((U * (n * (U_42_ - U))) / (Om * Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.5e-184], N[Sqrt[N[(N[(t + N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.4e+33], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.02e+239], N[Sqrt[N[(n * N[(2.0 * N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(N[(-2.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(U * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 2.50000000000000001e-184

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

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

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 54.8%

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

   if 2.50000000000000001e-184 < l < 3.3999999999999999e33

   1. Initial program 51.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 51.3%

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

   5. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

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

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

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

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

   7. Simplified 55.5%

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

   if 3.3999999999999999e33 < l < 1.02e239

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 64.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 76.8%

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

   8. Applied egg-rr 76.8%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

   10. Applied egg-rr 85.4%

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

   if 1.02e239 < l

   1. Initial program 23.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 65.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 65.0%

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

   6. Applied egg-rr 65.0%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

   9. Simplified 24.9%

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

   10. Taylor expanded in l around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 78.6%

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

   12. Simplified 78.6%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\left(\frac{\ell}{Om} \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)\right) \cdot \left(U \cdot \ell\right) + U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(-2 \cdot \frac{U}{Om} + \frac{U \cdot \left(n \cdot \left(U* - U\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.65 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{l\_m}{Om} \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;l\_m \leq 6.8 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\left(\frac{l\_m}{Om} \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)\right) \cdot \left(U \cdot l\_m\right) + U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.65e-184)
   (sqrt
    (* (+ t (* (* (/ l_m Om) (* n (/ l_m Om))) (- U* U))) (* n (* 2.0 U))))
   (if (<= l_m 6.8e+33)
     (sqrt
      (* (* 2.0 U) (* n (+ t (/ (* (* l_m l_m) (- (* U* (/ n Om)) 2.0)) Om)))))
     (sqrt
      (*
       n
       (*
        2.0
        (+
         (* (* (/ l_m Om) (+ -2.0 (/ n (/ Om (- U* U))))) (* U l_m))
         (* U t))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.65e-184) {
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else if (l_m <= 6.8e+33) {
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	} else {
		tmp = sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.65d-184) then
        tmp = sqrt(((t + (((l_m / om) * (n * (l_m / om))) * (u_42 - u))) * (n * (2.0d0 * u))))
    else if (l_m <= 6.8d+33) then
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((l_m * l_m) * ((u_42 * (n / om)) - 2.0d0)) / om)))))
    else
        tmp = sqrt((n * (2.0d0 * ((((l_m / om) * ((-2.0d0) + (n / (om / (u_42 - u))))) * (u * l_m)) + (u * t)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.65e-184) {
		tmp = Math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else if (l_m <= 6.8e+33) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	} else {
		tmp = Math.sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.65e-184:
		tmp = math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))))
	elif l_m <= 6.8e+33:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))))
	else:
		tmp = math.sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.65e-184)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(l_m / Om) * Float64(n * Float64(l_m / Om))) * Float64(U_42_ - U))) * Float64(n * Float64(2.0 * U))));
	elseif (l_m <= 6.8e+33)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(Float64(l_m * l_m) * Float64(Float64(U_42_ * Float64(n / Om)) - 2.0)) / Om)))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(Float64(Float64(Float64(l_m / Om) * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))) * Float64(U * l_m)) + Float64(U * t)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.65e-184)
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	elseif (l_m <= 6.8e+33)
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	else
		tmp = sqrt((n * (2.0 * ((((l_m / Om) * (-2.0 + (n / (Om / (U_42_ - U))))) * (U * l_m)) + (U * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.65e-184], N[Sqrt[N[(N[(t + N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 6.8e+33], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.6500000000000002e-184

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

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

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 54.8%

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

   if 2.6500000000000002e-184 < l < 6.7999999999999999e33

   1. Initial program 51.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 51.3%

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

   5. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

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

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

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

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

   7. Simplified 55.5%

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

   if 6.7999999999999999e33 < l

   1. Initial program 43.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 64.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 73.5%

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

   8. Applied egg-rr 73.5%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

   10. Applied egg-rr 77.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.65 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(\left(\frac{\ell}{Om} \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)\right) \cdot \left(U \cdot \ell\right) + U \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{if}\;Om \leq -9.5 \cdot 10^{-123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{n \cdot \frac{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)\right)}{Om}}{Om}}\\ \mathbf{elif}\;Om \leq 6.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot -2\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot U\right)\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0))))))))))
   (if (<= Om -9.5e-123)
     t_1
     (if (<= Om 4e-308)
       (sqrt (* n (/ (/ (* 2.0 (* U (* U* (* n (* l_m l_m))))) Om) Om)))
       (if (<= Om 6.6e-81)
         (/ (sqrt (* (* n -2.0) (* (- U U*) (* (* l_m l_m) (* n U))))) Om)
         t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	double tmp;
	if (Om <= -9.5e-123) {
		tmp = t_1;
	} else if (Om <= 4e-308) {
		tmp = sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)));
	} else if (Om <= 6.6e-81) {
		tmp = sqrt(((n * -2.0) * ((U - U_42_) * ((l_m * l_m) * (n * U))))) / Om;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
    if (om <= (-9.5d-123)) then
        tmp = t_1
    else if (om <= 4d-308) then
        tmp = sqrt((n * (((2.0d0 * (u * (u_42 * (n * (l_m * l_m))))) / om) / om)))
    else if (om <= 6.6d-81) then
        tmp = sqrt(((n * (-2.0d0)) * ((u - u_42) * ((l_m * l_m) * (n * u))))) / om
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	double tmp;
	if (Om <= -9.5e-123) {
		tmp = t_1;
	} else if (Om <= 4e-308) {
		tmp = Math.sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)));
	} else if (Om <= 6.6e-81) {
		tmp = Math.sqrt(((n * -2.0) * ((U - U_42_) * ((l_m * l_m) * (n * U))))) / Om;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))
	tmp = 0
	if Om <= -9.5e-123:
		tmp = t_1
	elif Om <= 4e-308:
		tmp = math.sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)))
	elif Om <= 6.6e-81:
		tmp = math.sqrt(((n * -2.0) * ((U - U_42_) * ((l_m * l_m) * (n * U))))) / Om
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))))
	tmp = 0.0
	if (Om <= -9.5e-123)
		tmp = t_1;
	elseif (Om <= 4e-308)
		tmp = sqrt(Float64(n * Float64(Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l_m * l_m))))) / Om) / Om)));
	elseif (Om <= 6.6e-81)
		tmp = Float64(sqrt(Float64(Float64(n * -2.0) * Float64(Float64(U - U_42_) * Float64(Float64(l_m * l_m) * Float64(n * U))))) / Om);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	tmp = 0.0;
	if (Om <= -9.5e-123)
		tmp = t_1;
	elseif (Om <= 4e-308)
		tmp = sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)));
	elseif (Om <= 6.6e-81)
		tmp = sqrt(((n * -2.0) * ((U - U_42_) * ((l_m * l_m) * (n * U))))) / Om;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -9.5e-123], t$95$1, If[LessEqual[Om, 4e-308], N[Sqrt[N[(n * N[(N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 6.6e-81], N[(N[Sqrt[N[(N[(n * -2.0), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -9.5000000000000002e-123 or 6.59999999999999975e-81 < Om

   1. Initial program 56.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 56.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 61.7%

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

   8. Applied egg-rr 61.7%

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

   9. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 56.4%

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

   11. Simplified 56.4%

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

   if -9.5000000000000002e-123 < Om < 4.00000000000000013e-308

   1. Initial program 32.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 53.7%

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

   5. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 42.7%

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

   7. Simplified 42.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. associate-*r/ N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

      16. --lowering--.f64 42.8%

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

   9. Applied egg-rr 42.8%

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

   10. Taylor expanded in U around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 47.0%

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

   12. Simplified 47.0%

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

   if 4.00000000000000013e-308 < Om < 6.59999999999999975e-81

   1. Initial program 40.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 42.8%

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

   5. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 34.3%

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

   7. Simplified 34.3%

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

      1. associate-*r/ N/A

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

      2. sqrt-div N/A

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

      3. pow2 N/A

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

      4. sqrt-pow1 N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

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

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

   9. Applied egg-rr 50.1%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.5 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{n \cdot \frac{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om}}{Om}}\\ \mathbf{elif}\;Om \leq 6.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot -2\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.2 \cdot 10^{+151} \lor \neg \left(Om \leq 1.8 \cdot 10^{+123}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -1.2e+151) (not (<= Om 1.8e+123)))
   (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0))))))))
   (sqrt
    (* (* 2.0 U) (* n (+ t (/ (* (* l_m l_m) (- (* U* (/ n Om)) 2.0)) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.2e+151) || !(Om <= 1.8e+123)) {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.2d+151)) .or. (.not. (om <= 1.8d+123))) then
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((l_m * l_m) * ((u_42 * (n / om)) - 2.0d0)) / om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.2e+151) || !(Om <= 1.8e+123)) {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -1.2e+151) or not (Om <= 1.8e+123):
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.2e+151) || !(Om <= 1.8e+123))
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(Float64(l_m * l_m) * Float64(Float64(U_42_ * Float64(n / Om)) - 2.0)) / Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.2e+151) || ~((Om <= 1.8e+123)))
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.2e+151], N[Not[LessEqual[Om, 1.8e+123]], $MachinePrecision]], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.20000000000000005e151 or 1.79999999999999999e123 < Om

   1. Initial program 61.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 59.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 70.4%

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

   8. Applied egg-rr 70.4%

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

   9. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 69.6%

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

   11. Simplified 69.6%

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

   if -1.20000000000000005e151 < Om < 1.79999999999999999e123

   1. Initial program 46.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 50.8%

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

   5. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

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

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

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

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

   7. Simplified 53.9%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.2 \cdot 10^{+151} \lor \neg \left(Om \leq 1.8 \cdot 10^{+123}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.7 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{l\_m}{Om} \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.7e-184)
   (sqrt
    (* (+ t (* (* (/ l_m Om) (* n (/ l_m Om))) (- U* U))) (* n (* 2.0 U))))
   (sqrt
    (*
     n
     (*
      2.0
      (*
       U
       (+
        t
        (* l_m (* (/ 1.0 Om) (* l_m (+ -2.0 (/ n (/ Om (- U* U))))))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e-184) {
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * (-2.0 + (n / (Om / (U_42_ - U))))))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.7d-184) then
        tmp = sqrt(((t + (((l_m / om) * (n * (l_m / om))) * (u_42 - u))) * (n * (2.0d0 * u))))
    else
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * ((-2.0d0) + (n / (om / (u_42 - u))))))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e-184) {
		tmp = Math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * (-2.0 + (n / (Om / (U_42_ - U))))))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.7e-184:
		tmp = math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))))
	else:
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * (-2.0 + (n / (Om / (U_42_ - U))))))))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.7e-184)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(l_m / Om) * Float64(n * Float64(l_m / Om))) * Float64(U_42_ - U))) * Float64(n * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.7e-184)
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	else
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * (-2.0 + (n / (Om / (U_42_ - U))))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.7e-184], N[Sqrt[N[(N[(t + N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7000000000000001e-184

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

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

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 54.8%

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

   if 2.7000000000000001e-184 < l

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 57.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 62.8%

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

   6. Applied egg-rr 62.8%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 62.9%

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

   8. Applied egg-rr 62.9%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -8 \cdot 10^{-28} \lor \neg \left(Om \leq 2.4 \cdot 10^{-81}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\left(l\_m \cdot -2\right) \cdot \left(l\_m \cdot \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -8e-28) (not (<= Om 2.4e-81)))
   (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0))))))))
   (sqrt (* n (/ (* (* l_m -2.0) (* l_m (/ (* U (* n (- U U*))) Om))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -8e-28) || !(Om <= 2.4e-81)) {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = sqrt((n * (((l_m * -2.0) * (l_m * ((U * (n * (U - U_42_))) / Om))) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-8d-28)) .or. (.not. (om <= 2.4d-81))) then
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
    else
        tmp = sqrt((n * (((l_m * (-2.0d0)) * (l_m * ((u * (n * (u - u_42))) / om))) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -8e-28) || !(Om <= 2.4e-81)) {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = Math.sqrt((n * (((l_m * -2.0) * (l_m * ((U * (n * (U - U_42_))) / Om))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -8e-28) or not (Om <= 2.4e-81):
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))
	else:
		tmp = math.sqrt((n * (((l_m * -2.0) * (l_m * ((U * (n * (U - U_42_))) / Om))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -8e-28) || !(Om <= 2.4e-81))
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(Float64(l_m * -2.0) * Float64(l_m * Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Om))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -8e-28) || ~((Om <= 2.4e-81)))
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	else
		tmp = sqrt((n * (((l_m * -2.0) * (l_m * ((U * (n * (U - U_42_))) / Om))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -8e-28], N[Not[LessEqual[Om, 2.4e-81]], $MachinePrecision]], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -7.99999999999999977e-28 or 2.3999999999999999e-81 < Om

   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 58.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 64.0%

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

   8. Applied egg-rr 64.0%

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

   9. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 60.0%

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

   11. Simplified 60.0%

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

   if -7.99999999999999977e-28 < Om < 2.3999999999999999e-81

   1. Initial program 38.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 44.7%

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

   5. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 35.7%

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

   7. Simplified 35.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. associate-*r/ N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

      16. --lowering--.f64 36.0%

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

   9. Applied egg-rr 36.0%

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

       1. associate-*l* N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. associate-*r/ N/A

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

       8. *-commutative N/A

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

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

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

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

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

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

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

       12. --lowering--.f64 41.2%

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

   11. Applied egg-rr 41.2%

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

   12. Taylor expanded in n around 0

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

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

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

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

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

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

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

       4. --lowering--.f64 44.0%

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

   14. Simplified 44.0%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8 \cdot 10^{-28} \lor \neg \left(Om \leq 2.4 \cdot 10^{-81}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\left(\ell \cdot -2\right) \cdot \left(\ell \cdot \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.42 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{l\_m}{Om} \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \left(l\_m \cdot \frac{l\_m}{Om}\right) \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.42e-184)
   (sqrt
    (* (+ t (* (* (/ l_m Om) (* n (/ l_m Om))) (- U* U))) (* n (* 2.0 U))))
   (sqrt
    (*
     n
     (*
      2.0
      (* U (+ t (* (* l_m (/ l_m Om)) (+ -2.0 (* n (/ (- U* U) Om)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.42e-184) {
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else {
		tmp = sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 + (n * ((U_42_ - U) / Om)))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.42d-184) then
        tmp = sqrt(((t + (((l_m / om) * (n * (l_m / om))) * (u_42 - u))) * (n * (2.0d0 * u))))
    else
        tmp = sqrt((n * (2.0d0 * (u * (t + ((l_m * (l_m / om)) * ((-2.0d0) + (n * ((u_42 - u) / om)))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.42e-184) {
		tmp = Math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 + (n * ((U_42_ - U) / Om)))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.42e-184:
		tmp = math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))))
	else:
		tmp = math.sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 + (n * ((U_42_ - U) / Om)))))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.42e-184)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(l_m / Om) * Float64(n * Float64(l_m / Om))) * Float64(U_42_ - U))) * Float64(n * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * Float64(-2.0 + Float64(n * Float64(Float64(U_42_ - U) / Om)))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.42e-184)
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	else
		tmp = sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 + (n * ((U_42_ - U) / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.42e-184], N[Sqrt[N[(N[(t + N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(-2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.41999999999999993e-184

   1. Initial program 54.3%

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

      1. *-commutative N/A

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

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

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in t around inf

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

   7. Simplified 54.8%

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

   if 1.41999999999999993e-184 < l

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

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

   3. Simplified 57.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 62.8%

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

   6. Applied egg-rr 62.8%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.42 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -9.5 \cdot 10^{-121} \lor \neg \left(Om \leq 4.4 \cdot 10^{-81}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)\right)}{Om}}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -9.5e-121) (not (<= Om 4.4e-81)))
   (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0))))))))
   (sqrt (* n (/ (/ (* 2.0 (* U (* U* (* n (* l_m l_m))))) Om) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -9.5e-121) || !(Om <= 4.4e-81)) {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-9.5d-121)) .or. (.not. (om <= 4.4d-81))) then
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
    else
        tmp = sqrt((n * (((2.0d0 * (u * (u_42 * (n * (l_m * l_m))))) / om) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -9.5e-121) || !(Om <= 4.4e-81)) {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = Math.sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -9.5e-121) or not (Om <= 4.4e-81):
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))
	else:
		tmp = math.sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -9.5e-121) || !(Om <= 4.4e-81))
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l_m * l_m))))) / Om) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -9.5e-121) || ~((Om <= 4.4e-81)))
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	else
		tmp = sqrt((n * (((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / Om) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -9.5e-121], N[Not[LessEqual[Om, 4.4e-81]], $MachinePrecision]], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -9.4999999999999994e-121 or 4.3999999999999998e-81 < Om

   1. Initial program 56.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. associate--l+ N/A

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

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 61.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.6%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{1}{Om}\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 61.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 61.7%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n}{\frac{Om}{U - U*}}\right)\right)\right)}\right)\right)\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 56.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 56.4%

       \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)} \]

   if -9.4999999999999994e-121 < Om < 4.3999999999999998e-81

   1. Initial program 36.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in Om around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\left(U \cdot {\ell}^{2}\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left(U \cdot {\ell}^{2}\right), \left(n \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right), \left(n \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right), \left(n \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(n \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right) \]

      12. *-lowering-*.f64 37.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \sqrt{n \cdot \color{blue}{\frac{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{\frac{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}}{Om}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\frac{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}\right), Om\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\frac{\left(-2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right), Om\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\left(-2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right), Om\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\left(-2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right), \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot U\right)\right), \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(-2 \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot \left(\ell \cdot \ell\right)\right), U\right), \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), U\right), \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), U\right), \left(n \cdot \frac{U - U*}{Om}\right)\right), Om\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), U\right), \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right), Om\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), U\right), \left(\frac{n}{\frac{Om}{U - U*}}\right)\right), Om\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), U\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right), Om\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), U\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right), Om\right)\right)\right) \]

      16. --lowering--.f64 38.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), U\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right), Om\right)\right)\right) \]

   9. Applied egg-rr 38.3%

      \[\leadsto \sqrt{n \cdot \color{blue}{\frac{\left(\left(-2 \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right) \cdot \frac{n}{\frac{Om}{U - U*}}}{Om}}} \]

   10. Taylor expanded in U around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)}, Om\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)}{Om}\right), Om\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

       8. *-lowering-*.f64 41.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), Om\right), Om\right)\right)\right) \]

   12. Simplified 41.8%

       \[\leadsto \sqrt{n \cdot \frac{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right)}{Om}}}{Om}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.5 \cdot 10^{-121} \lor \neg \left(Om \leq 4.4 \cdot 10^{-81}\right):\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om}}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 2.4 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{l\_m}{Om} \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 2.4e-281)
   (sqrt
    (* (+ t (* (* (/ l_m Om) (* n (/ l_m Om))) (- U* U))) (* n (* 2.0 U))))
   (sqrt
    (* (* 2.0 U) (* n (+ t (/ (* (* l_m l_m) (- (* U* (/ n Om)) 2.0)) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 2.4e-281) {
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 2.4d-281) then
        tmp = sqrt(((t + (((l_m / om) * (n * (l_m / om))) * (u_42 - u))) * (n * (2.0d0 * u))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((l_m * l_m) * ((u_42 * (n / om)) - 2.0d0)) / om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 2.4e-281) {
		tmp = Math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 2.4e-281:
		tmp = math.sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 2.4e-281)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(l_m / Om) * Float64(n * Float64(l_m / Om))) * Float64(U_42_ - U))) * Float64(n * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(Float64(l_m * l_m) * Float64(Float64(U_42_ * Float64(n / Om)) - 2.0)) / Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= 2.4e-281)
		tmp = sqrt(((t + (((l_m / Om) * (n * (l_m / Om))) * (U_42_ - U))) * (n * (2.0 * U))));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m * l_m) * ((U_42_ * (n / Om)) - 2.0)) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, 2.4e-281], N[Sqrt[N[(N[(t + N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 2.4e-281

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right), \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

   4. Applied egg-rr 53.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{-2}{\frac{Om}{\ell \cdot \ell}}\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), n\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 59.4%

      \[\leadsto \sqrt{\left(\color{blue}{t} - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)} \]

   if 2.4e-281 < U

   1. Initial program 51.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot 2\right), \left(n \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \left(n \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \mathsf{*.f64}\left(n, \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \mathsf{*.f64}\left(n, \left(t - \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(\frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right), Om\right)\right)\right)\right)\right) \]

   7. Simplified 59.1%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \left(2 - U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.4 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(t + \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(U* \cdot \frac{n}{Om} - 2\right)}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 5.2 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(U* \cdot \frac{n \cdot l\_m}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \left(l\_m \cdot \frac{l\_m}{Om}\right) \cdot \left(-2 - \frac{n \cdot U}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 5.2e+42)
   (sqrt
    (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* U* (/ (* n l_m) Om)))))))))
   (sqrt
    (* n (* 2.0 (* U (+ t (* (* l_m (/ l_m Om)) (- -2.0 (/ (* n U) Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 5.2e+42) {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))));
	} else {
		tmp = sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 - ((n * U) / Om))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 5.2d+42) then
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (u_42 * ((n * l_m) / om)))))))))
    else
        tmp = sqrt((n * (2.0d0 * (u * (t + ((l_m * (l_m / om)) * ((-2.0d0) - ((n * u) / om))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 5.2e+42) {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 - ((n * U) / Om))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= 5.2e+42:
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))))
	else:
		tmp = math.sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 - ((n * U) / Om))))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 5.2e+42)
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(U_42_ * Float64(Float64(n * l_m) / Om)))))))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * Float64(-2.0 - Float64(Float64(n * U) / Om))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= 5.2e+42)
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))));
	else
		tmp = sqrt((n * (2.0 * (U * (t + ((l_m * (l_m / Om)) * (-2.0 - ((n * U) / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, 5.2e+42], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(-2.0 - N[(N[(n * U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 5.1999999999999998e42

   1. Initial program 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. 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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 54.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 54.8%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{1}{Om}\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 55.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 55.4%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n}{\frac{Om}{U - U*}}\right)\right)\right)}\right)\right)\right)} \]

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 50.6%

       \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)\right)\right)\right)} \]

   if 5.1999999999999998e42 < Om

   1. Initial program 59.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 67.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 67.0%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]

   7. Taylor expanded in U around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \color{blue}{\left(\frac{U \cdot n}{Om}\right)}\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(U \cdot n\right), Om\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 61.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, n\right), Om\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 61.9%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \ell\right) \cdot \left(-2 - \color{blue}{\frac{U \cdot n}{Om}}\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5.2 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 - \frac{n \cdot U}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 5.1 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(U* \cdot \frac{n \cdot l\_m}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 5.1e+42)
   (sqrt
    (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* U* (/ (* n l_m) Om)))))))))
   (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 5.1e+42) {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))));
	} else {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 5.1d+42) then
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (u_42 * ((n * l_m) / om)))))))))
    else
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 5.1e+42) {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= 5.1e+42:
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))))
	else:
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 5.1e+42)
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(U_42_ * Float64(Float64(n * l_m) / Om)))))))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= 5.1e+42)
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (U_42_ * ((n * l_m) / Om)))))))));
	else
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, 5.1e+42], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 5.0999999999999999e42

   1. Initial program 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. 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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 54.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 54.8%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{1}{Om}\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 55.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 55.4%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n}{\frac{Om}{U - U*}}\right)\right)\right)}\right)\right)\right)} \]

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 50.6%

       \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)\right)\right)\right)} \]

   if 5.0999999999999999e42 < Om

   1. Initial program 59.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 67.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 67.0%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{1}{Om}\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 67.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n}{\frac{Om}{U - U*}}\right)\right)\right)}\right)\right)\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 61.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 61.4%

       \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5.1 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.85e+264)
   (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0))))))))
   (sqrt (* n (/ (* 2.0 (* U (* U* (* n (* l_m l_m))))) (* Om Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.85e+264) {
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.85d+264) then
        tmp = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * (u_42 * (n * (l_m * l_m))))) / (om * om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.85e+264) {
		tmp = Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.85e+264:
		tmp = math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.85e+264)
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l_m * l_m))))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.85e+264)
		tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
	else
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.85e+264], N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.85e264

   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 57.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 57.9%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{1}{Om}\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 58.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 58.4%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n}{\frac{Om}{U - U*}}\right)\right)\right)}\right)\right)\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 49.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 49.2%

       \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)} \]

   if 1.85e264 < l

   1. Initial program 23.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left(n \cdot {\ell}^{2}\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 63.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right) \]

   7. Simplified 63.8%

      \[\leadsto \sqrt{n \cdot \color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\\ \mathbf{if}\;U \leq -1.9 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (+ t (/ (* (* l_m l_m) -2.0) Om))))
   (if (<= U -1.9e-180)
     (sqrt (* (* U (* 2.0 n)) t_1))
     (sqrt (* 2.0 (* U (* n t_1)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double tmp;
	if (U <= -1.9e-180) {
		tmp = sqrt(((U * (2.0 * n)) * t_1));
	} else {
		tmp = sqrt((2.0 * (U * (n * t_1))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + (((l_m * l_m) * (-2.0d0)) / om)
    if (u <= (-1.9d-180)) then
        tmp = sqrt(((u * (2.0d0 * n)) * t_1))
    else
        tmp = sqrt((2.0d0 * (u * (n * t_1))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double tmp;
	if (U <= -1.9e-180) {
		tmp = Math.sqrt(((U * (2.0 * n)) * t_1));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t_1))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t + (((l_m * l_m) * -2.0) / Om)
	tmp = 0
	if U <= -1.9e-180:
		tmp = math.sqrt(((U * (2.0 * n)) * t_1))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t_1))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om))
	tmp = 0.0
	if (U <= -1.9e-180)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t_1));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_1))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = t + (((l_m * l_m) * -2.0) / Om);
	tmp = 0.0;
	if (U <= -1.9e-180)
		tmp = sqrt(((U * (2.0 * n)) * t_1));
	else
		tmp = sqrt((2.0 * (U * (n * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.9e-180], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -1.9e-180

   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. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 49.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right) \]

   5. Simplified 49.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}} \]

   if -1.9e-180 < U

   1. Initial program 48.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 56.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 56.2%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om} + U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) + -1 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) + \left(\mathsf{neg}\left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) - \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

   9. Simplified 44.0%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \frac{\frac{U \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}}{Om}\right)}\right)} \]

   10. Taylor expanded in n around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 45.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right)\right)\right) \]

   12. Simplified 45.1%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.9 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{+38}:\\ \;\;\;\;{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -7.5e+38)
   (pow (* n (* 2.0 (* U t))) 0.5)
   (sqrt (* 2.0 (* U (* n (+ t (/ (* (* l_m l_m) -2.0) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -7.5e+38) {
		tmp = pow((n * (2.0 * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (((l_m * l_m) * -2.0) / Om))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-7.5d+38)) then
        tmp = (n * (2.0d0 * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l_m * l_m) * (-2.0d0)) / om))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -7.5e+38) {
		tmp = Math.pow((n * (2.0 * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l_m * l_m) * -2.0) / Om))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -7.5e+38:
		tmp = math.pow((n * (2.0 * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l_m * l_m) * -2.0) / Om))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -7.5e+38)
		tmp = Float64(n * Float64(2.0 * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -7.5e+38)
		tmp = (n * (2.0 * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * (t + (((l_m * l_m) * -2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -7.5e+38], N[Power[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -7.4999999999999999e38

   1. Initial program 63.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.3%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 52.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, t\right)\right)\right)\right) \]

   7. Simplified 52.7%

      \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, t\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]

   if -7.4999999999999999e38 < U

   1. Initial program 49.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 58.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 58.1%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om} + U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) + -1 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) + \left(\mathsf{neg}\left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) - \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)\right)}{Om}\right)\right)\right)\right)\right) \]

   9. Simplified 43.5%

      \[\leadsto \sqrt{n \cdot \left(2 \cdot \color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \frac{\frac{U \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}}{Om}\right)}\right)} \]

   10. Taylor expanded in n around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right)\right)\right) \]

   12. Simplified 44.0%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{+38}:\\ \;\;\;\;{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + l\_m \cdot \left(\frac{1}{Om} \cdot \left(l\_m \cdot -2\right)\right)\right)\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (* n (* 2.0 (* U (+ t (* l_m (* (/ 1.0 Om) (* l_m -2.0)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((n * (2.0d0 * (u * (t + (l_m * ((1.0d0 / om) * (l_m * (-2.0d0)))))))))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(n * Float64(2.0 * Float64(U * Float64(t + Float64(l_m * Float64(Float64(1.0 / Om) * Float64(l_m * -2.0))))))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((n * (2.0 * (U * (t + (l_m * ((1.0 / Om) * (l_m * -2.0))))))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(n * N[(2.0 * N[(U * N[(t + N[(l$95$m * N[(N[(1.0 / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. 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. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 58.2%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \ell\right), \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 58.2%

   \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\ell \cdot \frac{1}{Om}\right) \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{1}{Om}\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \left(\ell \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(n \cdot \frac{1}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. un-div-inv N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n}{\frac{Om}{U - U*}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U - U*}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U - U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. --lowering--.f64 58.7%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 58.7%

   \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n}{\frac{Om}{U - U*}}\right)\right)\right)}\right)\right)\right)} \]

9. Taylor expanded in n around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 48.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Simplified 48.0%

    \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right)} \]

12. Final simplification 48.0%

    \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \ell \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot -2\right)\right)\right)\right)\right)} \]
13. Add Preprocessing

Alternative 17: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U* \leq 5 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U* 5e-233) (sqrt (* (* U (* 2.0 n)) t)) (sqrt (* U (* t (* 2.0 n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 5e-233) {
		tmp = sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = sqrt((U * (t * (2.0 * n))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 5d-233) then
        tmp = sqrt(((u * (2.0d0 * n)) * t))
    else
        tmp = sqrt((u * (t * (2.0d0 * n))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 5e-233) {
		tmp = Math.sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = Math.sqrt((U * (t * (2.0 * n))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U_42_ <= 5e-233:
		tmp = math.sqrt(((U * (2.0 * n)) * t))
	else:
		tmp = math.sqrt((U * (t * (2.0 * n))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 5e-233)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t));
	else
		tmp = sqrt(Float64(U * Float64(t * Float64(2.0 * n))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 5e-233)
		tmp = sqrt(((U * (2.0 * n)) * t));
	else
		tmp = sqrt((U * (t * (2.0 * n))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U$42$, 5e-233], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 5.00000000000000012e-233

   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. 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 36.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if 5.00000000000000012e-233 < U*

   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 41.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, t\right)\right)\right)\right) \]

   7. Simplified 41.8%

      \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot t\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(n \cdot 2\right) \cdot t\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(n \cdot 2\right) \cdot t\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 2\right), t\right), U\right)\right) \]

      6. *-lowering-*.f64 45.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right), U\right)\right) \]

   9. Applied egg-rr 45.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot 2\right) \cdot t\right) \cdot U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 5 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U* \leq 7.4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U* 7.4e-233)
   (sqrt (* (* U (* 2.0 n)) t))
   (sqrt (* (* 2.0 U) (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 7.4e-233) {
		tmp = sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 7.4d-233) then
        tmp = sqrt(((u * (2.0d0 * n)) * t))
    else
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 7.4e-233) {
		tmp = Math.sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U_42_ <= 7.4e-233:
		tmp = math.sqrt(((U * (2.0 * n)) * t))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 7.4e-233)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 7.4e-233)
		tmp = sqrt(((U * (2.0 * n)) * t));
	else
		tmp = sqrt(((2.0 * U) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U$42$, 7.4e-233], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 7.3999999999999996e-233

   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. 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 36.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if 7.3999999999999996e-233 < U*

   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. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      9. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot 2\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \left(n \cdot t\right)\right)\right) \]

      5. *-lowering-*.f64 45.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 45.0%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 7.4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ {\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* n (* 2.0 (* U t))) 0.5))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return pow((n * (2.0 * (U * t))), 0.5);
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (n * (2.0d0 * (u * t))) ** 0.5d0
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.pow((n * (2.0 * (U * t))), 0.5);
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.pow((n * (2.0 * (U * t))), 0.5)

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return Float64(n * Float64(2.0 * Float64(U * t))) ^ 0.5
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = (n * (2.0 * (U * t))) ^ 0.5;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Power[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. 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. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot t\right)\right)\right)\right) \]

   2. *-lowering-*.f64 38.2%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, t\right)\right)\right)\right) \]

7. Simplified 38.2%

   \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}} \]
8. Step-by-step derivation

   1. pow1/2 N/A

      \[\leadsto {\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

   5. *-lowering-*.f64 39.3%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, t\right)\right)\right), \frac{1}{2}\right) \]

9. Applied egg-rr 39.3%

   \[\leadsto \color{blue}{{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]
10. Add Preprocessing

Alternative 20: 35.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* n (* 2.0 (* U t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((n * (2.0 * (U * t))));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((n * (2.0d0 * (u * t))))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((n * (2.0 * (U * t))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((n * (2.0 * (U * t))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(n * Float64(2.0 * Float64(U * t))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((n * (2.0 * (U * t))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. 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. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot \left(-2 - n \cdot \frac{U - U*}{Om}\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \left(U \cdot t\right)\right)\right)\right) \]

   2. *-lowering-*.f64 38.2%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, t\right)\right)\right)\right) \]

7. Simplified 38.2%

   \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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*))))))