Toniolo and Linder, Equation (13)

Percentage Accurate: 49.7% → 64.6%

Time: 30.1s

Alternatives: 9

Speedup: 1.0×

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: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l_m \cdot \frac{l_m}{Om}\\ t_2 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot t_1\right) + t_2\right)\right)}\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(t_2 + \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om)))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ (+ t (* -2.0 t_1)) t_2))))
     (if (<= t_4 2e+150)
       (sqrt (* t_3 (+ t_2 (- t (* 2.0 t_1)))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 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 t_1 = l_m * (l_m / Om);
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	} else if (t_4 <= 2e+150) {
		tmp = sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((n / (pow(Om, 2.0) / (U_42_ - U))) - (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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = l_m * (l_m / om)
    t_2 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_3 = (2.0d0 * n) * u
    t_4 = sqrt((t_3 * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_2)))
    if (t_4 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t + ((-2.0d0) * t_1)) + t_2))))
    else if (t_4 <= 2d+150) then
        tmp = sqrt((t_3 * (t_2 + (t - (2.0d0 * t_1)))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * u) * ((n / ((om ** 2.0d0) / (u_42 - u))) - (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 t_1 = l_m * (l_m / Om);
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	} else if (t_4 <= 2e+150) {
		tmp = Math.sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * U) * ((n / (Math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	t_2 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))))
	elif t_4 <= 2e+150:
		tmp = math.sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * U) * ((n / (math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m * Float64(l_m / Om))
	t_2 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * t_1)) + t_2))));
	elseif (t_4 <= 2e+150)
		tmp = sqrt(Float64(t_3 * Float64(t_2 + Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(n / Float64((Om ^ 2.0) / Float64(U_42_ - U))) - Float64(2.0 / Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = l_m * (l_m / Om);
	t_2 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	elseif (t_4 <= 2e+150)
		tmp = sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((n / ((Om ^ 2.0) / (U_42_ - U))) - (2.0 / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e+150], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.1%

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

   3. Simplified 23.9%

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

      1. associate-*r/ 23.9%

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

      2. associate-*r* 33.9%

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

      3. cancel-sign-sub-inv 33.9%

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

      4. cancel-sign-sub-inv 33.9%

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

      5. metadata-eval 33.9%

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

      6. pow2 33.9%

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

   6. Applied egg-rr 33.9%

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

      1. unpow2 33.9%

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

      2. associate-*l/ 33.9%

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

   8. Applied egg-rr 33.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999996e150

   1. Initial program 95.5%

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

      1. associate-*l/ 95.6%

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

   4. Applied egg-rr 95.6%

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

   if 1.99999999999999996e150 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 16.9%

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

   3. Simplified 26.9%

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

      1. add-sqr-sqrt 17.7%

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

      2. pow2 17.7%

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

      3. sqrt-prod 16.8%

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

      4. unpow2 16.8%

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

      5. sqrt-prod 12.6%

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

      6. add-sqr-sqrt 18.9%

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

   6. Applied egg-rr 18.9%

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

   7. Taylor expanded in l around inf 25.7%

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

      1. *-commutative 25.7%

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

      2. associate-*r* 26.3%

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

      3. *-commutative 26.3%

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

      4. associate-/l* 25.9%

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

      5. associate-*r/ 25.9%

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

      6. metadata-eval 25.9%

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

   9. Simplified 25.9%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l_m \cdot \frac{l_m}{Om}\\ t_2 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot t_1\right) + t_2\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(t_2 + \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om)))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ (+ t (* -2.0 t_1)) t_2))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ t_2 (- t (* 2.0 t_1)))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 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 t_1 = l_m * (l_m / Om);
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((n / (pow(Om, 2.0) / (U_42_ - U))) - (2.0 / 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 = l_m * (l_m / Om);
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((n / (Math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	t_2 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * ((n / (math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.1%

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

   3. Simplified 23.9%

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

      1. associate-*r/ 23.9%

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

      2. associate-*r* 33.9%

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

      3. cancel-sign-sub-inv 33.9%

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

      4. cancel-sign-sub-inv 33.9%

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

      5. metadata-eval 33.9%

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

      6. pow2 33.9%

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

   6. Applied egg-rr 33.9%

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

      1. unpow2 33.9%

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

      2. associate-*l/ 33.9%

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

   8. Applied egg-rr 33.9%

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

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

   1. Initial program 59.0%

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

      1. associate-*l/ 66.3%

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

   4. Applied egg-rr 66.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in l around inf 38.2%

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

      1. associate-/l* 36.6%

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

      2. associate-*r/ 36.6%

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

      3. metadata-eval 36.6%

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

   7. Simplified 36.6%

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

4. Final simplification 57.8%

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

Alternative 3: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t_1 + \left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {l_m}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \end{array} \]

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 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 U)) (sqrt (* n t)))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ t_1 (- t (* 2.0 (* l_m (/ l_m Om)))))))
       (pow (* (/ (* n (pow l_m 2.0)) Om) (* U -4.0)) 0.5)))))

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 = (2.0 * n) * U;
	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 * U)) * sqrt((n * t));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (t_1 + (t - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = pow((((n * pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	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 = (2.0 * n) * U;
	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 * U)) * Math.sqrt((n * t));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * (t_1 + (t - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.pow((((n * Math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	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 = (2.0 * n) * U
	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 * U)) * math.sqrt((n * t))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * (t_1 + (t - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.pow((((n * math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5)
	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(Float64(2.0 * n) * U)
	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(2.0 * U)) * sqrt(Float64(n * t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(t_1 + Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(Float64(n * (l_m ^ 2.0)) / Om) * Float64(U * -4.0)) ^ 0.5;
	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 = (2.0 * n) * U;
	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 * U)) * sqrt((n * t));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * (t_1 + (t - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (((n * (l_m ^ 2.0)) / Om) * (U * -4.0)) ^ 0.5;
	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[(N[(2.0 * n), $MachinePrecision] * U), $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[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(t$95$1 + N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.1%

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

   3. Simplified 12.7%

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

   5. Taylor expanded in l around 0 17.4%

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

      1. pow1/2 17.4%

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

      2. associate-*r* 17.5%

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

      3. unpow-prod-down 32.4%

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

      4. pow1/2 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. unpow1/2 32.4%

         \[\leadsto \color{blue}{\sqrt{2 \cdot U}} \cdot \sqrt{n \cdot t} \]

   9. Simplified 32.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}} \]

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

   1. Initial program 59.0%

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

      1. associate-*l/ 66.3%

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

   4. Applied egg-rr 66.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in l around inf 36.0%

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

      1. associate-*r* 36.0%

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

      2. associate-*r* 36.2%

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

      3. *-commutative 36.2%

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

      4. associate-*r/ 36.2%

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

      5. metadata-eval 36.2%

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

   7. Simplified 36.2%

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

   8. Taylor expanded in n around 0 4.5%

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

      1. pow1/2 36.3%

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

      2. associate-*r* 36.3%

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

      3. unpow-prod-down 21.8%

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

      4. *-commutative 21.8%

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

      5. pow1/2 2.1%

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

      6. associate-/l* 2.1%

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

   10. Applied egg-rr 2.1%

       \[\leadsto \color{blue}{{\left(\left(U \cdot -2\right) \cdot 2\right)}^{0.5} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}}} \]
   11. Step-by-step derivation

       1. unpow1/2 2.1%

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

       2. associate-*l* 2.1%

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

       3. metadata-eval 2.1%

          \[\leadsto \sqrt{U \cdot \color{blue}{-4}} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}} \]

       4. associate-/r/ 2.1%

          \[\leadsto \sqrt{U \cdot -4} \cdot \sqrt{\color{blue}{\frac{{\ell}^{2}}{Om} \cdot n}} \]

   12. Simplified 2.1%

       \[\leadsto \color{blue}{\sqrt{U \cdot -4} \cdot \sqrt{\frac{{\ell}^{2}}{Om} \cdot n}} \]
   13. Step-by-step derivation

       1. *-commutative 2.1%

          \[\leadsto \color{blue}{\sqrt{\frac{{\ell}^{2}}{Om} \cdot n} \cdot \sqrt{U \cdot -4}} \]

       2. pow1/2 22.2%

          \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{0.5}} \cdot \sqrt{U \cdot -4} \]

       3. metadata-eval 22.2%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{U \cdot -4} \]

       4. pow1/2 22.2%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(U \cdot -4\right)}^{0.5}} \]

       5. metadata-eval 22.2%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(U \cdot -4\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-prod-down 36.2%

          \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{2}}{Om} \cdot n\right) \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       7. associate-*l/ 36.3%

          \[\leadsto {\left(\color{blue}{\frac{{\ell}^{2} \cdot n}{Om}} \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       8. metadata-eval 36.3%

          \[\leadsto {\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{\color{blue}{0.5}} \]

   14. Applied egg-rr 36.3%

       \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {\ell}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l_m \cdot \frac{l_m}{Om}\\ t_2 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot t_1\right) + t_2\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(t_2 + \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {l_m}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om)))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ (+ t (* -2.0 t_1)) t_2))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ t_2 (- t (* 2.0 t_1)))))
       (pow (* (/ (* n (pow l_m 2.0)) Om) (* U -4.0)) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	} else {
		tmp = pow((((n * pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	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 = l_m * (l_m / Om);
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	} else {
		tmp = Math.pow((((n * Math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	t_2 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))))
	else:
		tmp = math.pow((((n * math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m * Float64(l_m / Om))
	t_2 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * t_1)) + t_2))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(t_2 + Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = Float64(Float64(Float64(n * (l_m ^ 2.0)) / Om) * Float64(U * -4.0)) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = l_m * (l_m / Om);
	t_2 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = sqrt((t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + t_2))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * (t_2 + (t - (2.0 * t_1)))));
	else
		tmp = (((n * (l_m ^ 2.0)) / Om) * (U * -4.0)) ^ 0.5;
	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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.1%

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

   3. Simplified 23.9%

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

      1. associate-*r/ 23.9%

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

      2. associate-*r* 33.9%

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

      3. cancel-sign-sub-inv 33.9%

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

      4. cancel-sign-sub-inv 33.9%

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

      5. metadata-eval 33.9%

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

      6. pow2 33.9%

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

   6. Applied egg-rr 33.9%

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

      1. unpow2 33.9%

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

      2. associate-*l/ 33.9%

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

   8. Applied egg-rr 33.9%

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

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

   1. Initial program 59.0%

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

      1. associate-*l/ 66.3%

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

   4. Applied egg-rr 66.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in l around inf 36.0%

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

      1. associate-*r* 36.0%

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

      2. associate-*r* 36.2%

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

      3. *-commutative 36.2%

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

      4. associate-*r/ 36.2%

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

      5. metadata-eval 36.2%

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

   7. Simplified 36.2%

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

   8. Taylor expanded in n around 0 4.5%

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

      1. pow1/2 36.3%

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

      2. associate-*r* 36.3%

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

      3. unpow-prod-down 21.8%

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

      4. *-commutative 21.8%

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

      5. pow1/2 2.1%

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

      6. associate-/l* 2.1%

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

   10. Applied egg-rr 2.1%

       \[\leadsto \color{blue}{{\left(\left(U \cdot -2\right) \cdot 2\right)}^{0.5} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}}} \]
   11. Step-by-step derivation

       1. unpow1/2 2.1%

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

       2. associate-*l* 2.1%

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

       3. metadata-eval 2.1%

          \[\leadsto \sqrt{U \cdot \color{blue}{-4}} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}} \]

       4. associate-/r/ 2.1%

          \[\leadsto \sqrt{U \cdot -4} \cdot \sqrt{\color{blue}{\frac{{\ell}^{2}}{Om} \cdot n}} \]

   12. Simplified 2.1%

       \[\leadsto \color{blue}{\sqrt{U \cdot -4} \cdot \sqrt{\frac{{\ell}^{2}}{Om} \cdot n}} \]
   13. Step-by-step derivation

       1. *-commutative 2.1%

          \[\leadsto \color{blue}{\sqrt{\frac{{\ell}^{2}}{Om} \cdot n} \cdot \sqrt{U \cdot -4}} \]

       2. pow1/2 22.2%

          \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{0.5}} \cdot \sqrt{U \cdot -4} \]

       3. metadata-eval 22.2%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{U \cdot -4} \]

       4. pow1/2 22.2%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(U \cdot -4\right)}^{0.5}} \]

       5. metadata-eval 22.2%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(U \cdot -4\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-prod-down 36.2%

          \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{2}}{Om} \cdot n\right) \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       7. associate-*l/ 36.3%

          \[\leadsto {\left(\color{blue}{\frac{{\ell}^{2} \cdot n}{Om}} \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       8. metadata-eval 36.3%

          \[\leadsto {\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{\color{blue}{0.5}} \]

   14. Applied egg-rr 36.3%

       \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {\ell}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 8.8 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(n \cdot \left({\left(\frac{l_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - \left(\frac{2 \cdot \left(l_m \cdot l_m\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;l_m \leq 1.42 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + -2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {l_m}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8.8e+139)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (-
       (* n (* (pow (/ l_m Om) 2.0) (- U* U)))
       (- (/ (* 2.0 (* l_m l_m)) Om) t)))))
   (if (<= l_m 1.42e+231)
     (sqrt (* 2.0 (* (+ t (* -2.0 (* l_m (/ l_m Om)))) (* n U))))
     (pow (* (/ (* n (pow l_m 2.0)) Om) (* U -4.0)) 0.5))))

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 <= 8.8e+139) {
		tmp = sqrt(((2.0 * n) * (U * ((n * (pow((l_m / Om), 2.0) * (U_42_ - U))) - (((2.0 * (l_m * l_m)) / Om) - t)))));
	} else if (l_m <= 1.42e+231) {
		tmp = sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))));
	} else {
		tmp = pow((((n * pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	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 <= 8.8d+139) then
        tmp = sqrt(((2.0d0 * n) * (u * ((n * (((l_m / om) ** 2.0d0) * (u_42 - u))) - (((2.0d0 * (l_m * l_m)) / om) - t)))))
    else if (l_m <= 1.42d+231) then
        tmp = sqrt((2.0d0 * ((t + ((-2.0d0) * (l_m * (l_m / om)))) * (n * u))))
    else
        tmp = (((n * (l_m ** 2.0d0)) / om) * (u * (-4.0d0))) ** 0.5d0
    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 <= 8.8e+139) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((n * (Math.pow((l_m / Om), 2.0) * (U_42_ - U))) - (((2.0 * (l_m * l_m)) / Om) - t)))));
	} else if (l_m <= 1.42e+231) {
		tmp = Math.sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))));
	} else {
		tmp = Math.pow((((n * Math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 8.8e+139:
		tmp = math.sqrt(((2.0 * n) * (U * ((n * (math.pow((l_m / Om), 2.0) * (U_42_ - U))) - (((2.0 * (l_m * l_m)) / Om) - t)))))
	elif l_m <= 1.42e+231:
		tmp = math.sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))))
	else:
		tmp = math.pow((((n * math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8.8e+139)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * Float64(U_42_ - U))) - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - t)))));
	elseif (l_m <= 1.42e+231)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t + Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))) * Float64(n * U))));
	else
		tmp = Float64(Float64(Float64(n * (l_m ^ 2.0)) / Om) * Float64(U * -4.0)) ^ 0.5;
	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 <= 8.8e+139)
		tmp = sqrt(((2.0 * n) * (U * ((n * (((l_m / Om) ^ 2.0) * (U_42_ - U))) - (((2.0 * (l_m * l_m)) / Om) - t)))));
	elseif (l_m <= 1.42e+231)
		tmp = sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))));
	else
		tmp = (((n * (l_m ^ 2.0)) / Om) * (U * -4.0)) ^ 0.5;
	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, 8.8e+139], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.42e+231], N[Sqrt[N[(2.0 * N[(N[(t + N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.7999999999999998e139

   1. Initial program 50.1%

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

   3. Simplified 48.8%

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

   if 8.7999999999999998e139 < l < 1.42e231

   1. Initial program 1.7%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in n around 0 2.6%

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

      1. unpow2 1.8%

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

      2. associate-*l/ 30.1%

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

   7. Applied egg-rr 38.7%

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

   if 1.42e231 < l

   1. Initial program 26.4%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in l around inf 46.1%

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

      1. associate-*r* 46.1%

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

      2. associate-*r* 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

      5. metadata-eval 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in n around 0 27.1%

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

      1. pow1/2 46.8%

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

      2. associate-*r* 46.8%

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

      3. unpow-prod-down 21.1%

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

      4. *-commutative 21.1%

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

      5. pow1/2 7.6%

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

      6. associate-/l* 7.1%

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

   10. Applied egg-rr 7.1%

       \[\leadsto \color{blue}{{\left(\left(U \cdot -2\right) \cdot 2\right)}^{0.5} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}}} \]
   11. Step-by-step derivation

       1. unpow1/2 7.1%

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

       2. associate-*l* 7.1%

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

       3. metadata-eval 7.1%

          \[\leadsto \sqrt{U \cdot \color{blue}{-4}} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}} \]

       4. associate-/r/ 7.6%

          \[\leadsto \sqrt{U \cdot -4} \cdot \sqrt{\color{blue}{\frac{{\ell}^{2}}{Om} \cdot n}} \]

   12. Simplified 7.6%

       \[\leadsto \color{blue}{\sqrt{U \cdot -4} \cdot \sqrt{\frac{{\ell}^{2}}{Om} \cdot n}} \]
   13. Step-by-step derivation

       1. *-commutative 7.6%

          \[\leadsto \color{blue}{\sqrt{\frac{{\ell}^{2}}{Om} \cdot n} \cdot \sqrt{U \cdot -4}} \]

       2. pow1/2 21.1%

          \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{0.5}} \cdot \sqrt{U \cdot -4} \]

       3. metadata-eval 21.1%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{U \cdot -4} \]

       4. pow1/2 21.1%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(U \cdot -4\right)}^{0.5}} \]

       5. metadata-eval 21.1%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(U \cdot -4\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-prod-down 46.8%

          \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{2}}{Om} \cdot n\right) \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       7. associate-*l/ 46.8%

          \[\leadsto {\left(\color{blue}{\frac{{\ell}^{2} \cdot n}{Om}} \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       8. metadata-eval 46.8%

          \[\leadsto {\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{\color{blue}{0.5}} \]

   14. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {\ell}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 6.8 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + -2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {l_m}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.8e+230)
   (sqrt (* 2.0 (* (+ t (* -2.0 (* l_m (/ l_m Om)))) (* n U))))
   (pow (* (/ (* n (pow l_m 2.0)) Om) (* U -4.0)) 0.5)))

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 <= 6.8e+230) {
		tmp = sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))));
	} else {
		tmp = pow((((n * pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	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 <= 6.8d+230) then
        tmp = sqrt((2.0d0 * ((t + ((-2.0d0) * (l_m * (l_m / om)))) * (n * u))))
    else
        tmp = (((n * (l_m ** 2.0d0)) / om) * (u * (-4.0d0))) ** 0.5d0
    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 <= 6.8e+230) {
		tmp = Math.sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))));
	} else {
		tmp = Math.pow((((n * Math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 6.8e+230:
		tmp = math.sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))))
	else:
		tmp = math.pow((((n * math.pow(l_m, 2.0)) / Om) * (U * -4.0)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.8e+230)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t + Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))) * Float64(n * U))));
	else
		tmp = Float64(Float64(Float64(n * (l_m ^ 2.0)) / Om) * Float64(U * -4.0)) ^ 0.5;
	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 <= 6.8e+230)
		tmp = sqrt((2.0 * ((t + (-2.0 * (l_m * (l_m / Om)))) * (n * U))));
	else
		tmp = (((n * (l_m ^ 2.0)) / Om) * (U * -4.0)) ^ 0.5;
	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, 6.8e+230], N[Sqrt[N[(2.0 * N[(N[(t + N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.79999999999999973e230

   1. Initial program 45.3%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in n around 0 38.3%

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

      1. unpow2 46.5%

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

      2. associate-*l/ 51.3%

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

   7. Applied egg-rr 43.5%

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

   if 6.79999999999999973e230 < l

   1. Initial program 26.4%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in l around inf 46.1%

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

      1. associate-*r* 46.1%

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

      2. associate-*r* 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

      5. metadata-eval 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in n around 0 27.1%

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

      1. pow1/2 46.8%

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

      2. associate-*r* 46.8%

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

      3. unpow-prod-down 21.1%

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

      4. *-commutative 21.1%

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

      5. pow1/2 7.6%

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

      6. associate-/l* 7.1%

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

   10. Applied egg-rr 7.1%

       \[\leadsto \color{blue}{{\left(\left(U \cdot -2\right) \cdot 2\right)}^{0.5} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}}} \]
   11. Step-by-step derivation

       1. unpow1/2 7.1%

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

       2. associate-*l* 7.1%

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

       3. metadata-eval 7.1%

          \[\leadsto \sqrt{U \cdot \color{blue}{-4}} \cdot \sqrt{\frac{{\ell}^{2}}{\frac{Om}{n}}} \]

       4. associate-/r/ 7.6%

          \[\leadsto \sqrt{U \cdot -4} \cdot \sqrt{\color{blue}{\frac{{\ell}^{2}}{Om} \cdot n}} \]

   12. Simplified 7.6%

       \[\leadsto \color{blue}{\sqrt{U \cdot -4} \cdot \sqrt{\frac{{\ell}^{2}}{Om} \cdot n}} \]
   13. Step-by-step derivation

       1. *-commutative 7.6%

          \[\leadsto \color{blue}{\sqrt{\frac{{\ell}^{2}}{Om} \cdot n} \cdot \sqrt{U \cdot -4}} \]

       2. pow1/2 21.1%

          \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{0.5}} \cdot \sqrt{U \cdot -4} \]

       3. metadata-eval 21.1%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{U \cdot -4} \]

       4. pow1/2 21.1%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(U \cdot -4\right)}^{0.5}} \]

       5. metadata-eval 21.1%

          \[\leadsto {\left(\frac{{\ell}^{2}}{Om} \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(U \cdot -4\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-prod-down 46.8%

          \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{2}}{Om} \cdot n\right) \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       7. associate-*l/ 46.8%

          \[\leadsto {\left(\color{blue}{\frac{{\ell}^{2} \cdot n}{Om}} \cdot \left(U \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       8. metadata-eval 46.8%

          \[\leadsto {\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{\color{blue}{0.5}} \]

   14. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{{\left(\frac{{\ell}^{2} \cdot n}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{n \cdot {\ell}^{2}}{Om} \cdot \left(U \cdot -4\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.6% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (* 2.0 (* (+ t (* -2.0 (* l_m (/ l_m Om)))) (* n U)))))

C

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

3. Simplified 48.4%

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

5. Taylor expanded in n around 0 37.6%

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

   1. unpow2 45.2%

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

   2. associate-*l/ 50.2%

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

7. Applied egg-rr 42.9%

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

8. Final simplification 42.9%

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

Alternative 8: 37.1% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* 2.0 (* t (* n U))) 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((2.0 * (t * (n * U))), 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 = (2.0d0 * (t * (n * u))) ** 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((2.0 * (t * (n * U))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 44.1%

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

3. Simplified 48.4%

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

5. Taylor expanded in l around 0 30.8%

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

   1. pow1/2 31.3%

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

   2. associate-*r* 30.9%

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

   3. *-commutative 30.9%

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

7. Applied egg-rr 30.9%

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

8. Final simplification 30.9%

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

Alternative 9: 35.5% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (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_) {
	return sqrt((2.0 * (U * (n * 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((2.0d0 * (u * (n * 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((2.0 * (U * (n * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 44.1%

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

3. Simplified 48.4%

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

5. Taylor expanded in l around 0 30.8%

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

6. Final simplification 30.8%

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

Reproduce

herbie shell --seed 2024019 
(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*))))))