Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 64.9%

Time: 25.0s

Alternatives: 19

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% 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: 64.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 10^{-265}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{U \cdot \left(U* \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)\right) - 2 \cdot \left(U \cdot {l\_m}^{2}\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_2 1e-265)
     (sqrt
      (*
       (* 2.0 n)
       (+
        (/
         (-
          (* U (* U* (* (pow l_m 2.0) (/ n Om))))
          (* 2.0 (* U (pow l_m 2.0))))
         Om)
        (* U t))))
     (if (<= t_2 1e+305)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (- (* U* (/ n (pow Om 2.0))) (/ 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 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 1e-265) {
		tmp = sqrt(((2.0 * n) * ((((U * (U_42_ * (pow(l_m, 2.0) * (n / Om)))) - (2.0 * (U * pow(l_m, 2.0)))) / Om) + (U * t))));
	} else if (t_2 <= 1e+305) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((U_42_ * (n / pow(Om, 2.0))) - (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) :: tmp
    t_1 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_2 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_1)
    if (t_2 <= 1d-265) then
        tmp = sqrt(((2.0d0 * n) * ((((u * (u_42 * ((l_m ** 2.0d0) * (n / om)))) - (2.0d0 * (u * (l_m ** 2.0d0)))) / om) + (u * t))))
    else if (t_2 <= 1d+305) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_1 - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * u) * ((u_42 * (n / (om ** 2.0d0))) - (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 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 1e-265) {
		tmp = Math.sqrt(((2.0 * n) * ((((U * (U_42_ * (Math.pow(l_m, 2.0) * (n / Om)))) - (2.0 * (U * Math.pow(l_m, 2.0)))) / Om) + (U * t))));
	} else if (t_2 <= 1e+305) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * U) * ((U_42_ * (n / Math.pow(Om, 2.0))) - (2.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 1e-265:
		tmp = math.sqrt(((2.0 * n) * ((((U * (U_42_ * (math.pow(l_m, 2.0) * (n / Om)))) - (2.0 * (U * math.pow(l_m, 2.0)))) / Om) + (U * t))))
	elif t_2 <= 1e+305:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * U) * ((U_42_ * (n / math.pow(Om, 2.0))) - (2.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_2 <= 1e-265)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(Float64(Float64(U * Float64(U_42_ * Float64((l_m ^ 2.0) * Float64(n / Om)))) - Float64(2.0 * Float64(U * (l_m ^ 2.0)))) / Om) + Float64(U * t))));
	elseif (t_2 <= 1e+305)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - 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 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 1e-265)
		tmp = sqrt(((2.0 * n) * ((((U * (U_42_ * ((l_m ^ 2.0) * (n / Om)))) - (2.0 * (U * (l_m ^ 2.0)))) / Om) + (U * t))));
	elseif (t_2 <= 1e+305)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((U_42_ * (n / (Om ^ 2.0))) - (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[(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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-265], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(N[(U * N[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(U * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+305], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $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[(U$42$ * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999985e-266

   1. Initial program 20.9%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in Om around -inf 50.3%

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

   6. Taylor expanded in U around 0 50.3%

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

      1. mul-1-neg 50.3%

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

      2. associate-/l* 50.3%

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

      3. associate-/l* 50.3%

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

      4. associate-/l* 52.6%

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

   8. Simplified 52.6%

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

   if 9.99999999999999985e-266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.9999999999999994e304

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   if 9.9999999999999994e304 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 16.6%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in Om around -inf 21.9%

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

   6. Taylor expanded in U around 0 28.9%

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

      1. mul-1-neg 28.9%

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

      2. associate-/l* 28.9%

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

      3. associate-/l* 26.3%

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

      4. associate-/l* 25.3%

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

   8. Simplified 25.3%

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

   9. Taylor expanded in U around 0 24.6%

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

       1. mul-1-neg 24.6%

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

       2. unsub-neg 24.6%

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

       3. associate-*r* 24.6%

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

       4. associate-*r/ 22.7%

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

       5. *-commutative 22.7%

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

   11. Simplified 22.7%

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

   12. Taylor expanded in l around inf 22.3%

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

       1. *-commutative 22.3%

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

       2. associate-*r* 23.5%

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

       3. *-commutative 23.5%

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

       4. associate-/l* 23.5%

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

       5. associate-*r/ 23.5%

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

       6. metadata-eval 23.5%

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

   14. Simplified 23.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-265}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right) - 2 \cdot \left(U \cdot {\ell}^{2}\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 - \left(\frac{2 \cdot {l\_m}^{2}}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 5e-133)
     (sqrt (* (* 2.0 n) (* U (- t_1 (- (/ (* 2.0 (pow l_m 2.0)) Om) t)))))
     (if (<= t_2 4e+152)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (- (* U* (/ n (pow Om 2.0))) (/ 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 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 - (((2.0 * pow(l_m, 2.0)) / Om) - t)))));
	} else if (t_2 <= 4e+152) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((U_42_ * (n / pow(Om, 2.0))) - (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) :: tmp
    t_1 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_2 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_1)))
    if (t_2 <= 5d-133) then
        tmp = sqrt(((2.0d0 * n) * (u * (t_1 - (((2.0d0 * (l_m ** 2.0d0)) / om) - t)))))
    else if (t_2 <= 4d+152) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_1 - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * u) * ((u_42 * (n / (om ** 2.0d0))) - (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 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 - (((2.0 * Math.pow(l_m, 2.0)) / Om) - t)))));
	} else if (t_2 <= 4e+152) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * U) * ((U_42_ * (n / Math.pow(Om, 2.0))) - (2.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 5e-133:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 - (((2.0 * math.pow(l_m, 2.0)) / Om) - t)))))
	elif t_2 <= 4e+152:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * U) * ((U_42_ * (n / math.pow(Om, 2.0))) - (2.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 5e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 - Float64(Float64(Float64(2.0 * (l_m ^ 2.0)) / Om) - t)))));
	elseif (t_2 <= 4e+152)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - 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 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 5e-133)
		tmp = sqrt(((2.0 * n) * (U * (t_1 - (((2.0 * (l_m ^ 2.0)) / Om) - t)))));
	elseif (t_2 <= 4e+152)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((U_42_ * (n / (Om ^ 2.0))) - (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[(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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * 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$2, 5e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 - N[(N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+152], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $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[(U$42$ * N[(n / N[Power[Om, 2.0], $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 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999999e-133

   1. Initial program 25.0%

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

   3. Simplified 49.0%

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

      1. fma-undefine 49.0%

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

      2. associate-*r* 54.4%

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

      3. associate--r+ 54.4%

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

      4. *-commutative 54.4%

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

      5. cancel-sign-sub-inv 54.4%

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

      6. associate-*r/ 54.4%

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

      7. associate-*r/ 54.4%

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

      8. pow2 54.4%

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

   6. Applied egg-rr 54.4%

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

   if 4.9999999999999999e-133 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.0000000000000002e152

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   if 4.0000000000000002e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 15.6%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in Om around -inf 23.1%

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

   6. Taylor expanded in U around 0 29.7%

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

      1. mul-1-neg 29.7%

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

      2. associate-/l* 29.7%

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

      3. associate-/l* 27.3%

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

      4. associate-/l* 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in U around 0 25.7%

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

       1. mul-1-neg 25.7%

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

       2. unsub-neg 25.7%

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

       3. associate-*r* 24.8%

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

       4. associate-*r/ 23.9%

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

       5. *-commutative 23.9%

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

   11. Simplified 23.9%

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

   12. Taylor expanded in l around inf 21.9%

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

       1. *-commutative 21.9%

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

       2. associate-*r* 23.0%

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

       3. *-commutative 23.0%

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

       4. associate-/l* 23.0%

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

       5. associate-*r/ 23.0%

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

       6. metadata-eval 23.0%

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

   14. Simplified 23.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(\frac{2 \cdot {\ell}^{2}}{Om} - t\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 5e-133)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_2 4e+152)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (- (* U* (/ n (pow Om 2.0))) (/ 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 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= 4e+152) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((U_42_ * (n / pow(Om, 2.0))) - (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) :: tmp
    t_1 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_2 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_1)))
    if (t_2 <= 5d-133) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else if (t_2 <= 4d+152) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_1 - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * u) * ((u_42 * (n / (om ** 2.0d0))) - (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 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= 4e+152) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * U) * ((U_42_ * (n / Math.pow(Om, 2.0))) - (2.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 5e-133:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif t_2 <= 4e+152:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * U) * ((U_42_ * (n / math.pow(Om, 2.0))) - (2.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 5e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= 4e+152)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - 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 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 5e-133)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= 4e+152)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((U_42_ * (n / (Om ^ 2.0))) - (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[(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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * 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$2, 5e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+152], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $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[(U$42$ * N[(n / N[Power[Om, 2.0], $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 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999999e-133

   1. Initial program 25.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in n around 0 51.7%

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

   if 4.9999999999999999e-133 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.0000000000000002e152

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   if 4.0000000000000002e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 15.6%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in Om around -inf 23.1%

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

   6. Taylor expanded in U around 0 29.7%

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

      1. mul-1-neg 29.7%

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

      2. associate-/l* 29.7%

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

      3. associate-/l* 27.3%

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

      4. associate-/l* 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in U around 0 25.7%

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

       1. mul-1-neg 25.7%

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

       2. unsub-neg 25.7%

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

       3. associate-*r* 24.8%

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

       4. associate-*r/ 23.9%

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

       5. *-commutative 23.9%

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

   11. Simplified 23.9%

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

   12. Taylor expanded in l around inf 21.9%

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

       1. *-commutative 21.9%

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

       2. associate-*r* 23.0%

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

       3. *-commutative 23.0%

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

       4. associate-/l* 23.0%

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

       5. associate-*r/ 23.0%

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

       6. metadata-eval 23.0%

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

   14. Simplified 23.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 5e-133)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (* U* (/ n (pow Om 2.0))) (/ 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 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / pow(Om, 2.0))) - (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 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((U_42_ * (n / Math.pow(Om, 2.0))) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 5e-133:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * ((U_42_ * (n / math.pow(Om, 2.0))) - (2.0 / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 5e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - 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 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 5e-133)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / (Om ^ 2.0))) - (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[(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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * 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$2, 5e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(U$42$ * N[(n / N[Power[Om, 2.0], $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 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999999e-133

   1. Initial program 25.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in n around 0 51.7%

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

   if 4.9999999999999999e-133 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 67.0%

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

   3. Simplified 70.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 14.4%

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

   5. Taylor expanded in Om around -inf 23.6%

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

   6. Taylor expanded in U around 0 24.1%

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

      1. mul-1-neg 24.1%

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

      2. associate-/l* 24.1%

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

      3. associate-/l* 19.5%

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

      4. associate-/l* 21.7%

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

   8. Simplified 21.7%

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

   9. Taylor expanded in U around 0 19.7%

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

       1. mul-1-neg 19.7%

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

       2. unsub-neg 19.7%

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

       3. associate-*r* 17.4%

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

       4. associate-*r/ 19.6%

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

       5. *-commutative 19.6%

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

   11. Simplified 19.6%

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

   12. Taylor expanded in l around inf 38.7%

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

       1. *-commutative 38.7%

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

       2. associate-/l* 38.7%

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

       3. associate-*r/ 38.7%

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

       4. metadata-eval 38.7%

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

   14. Simplified 38.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot {l\_m}^{2}\right) \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

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
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 5e-133)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (sqrt
        (*
         -2.0
         (/ (* (* n (pow l_m 2.0)) (* U (- 2.0 (* U* (/ n Om))))) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt((-2.0 * (((n * pow(l_m, 2.0)) * (U * (2.0 - (U_42_ * (n / Om))))) / Om)));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 5e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (((n * Math.pow(l_m, 2.0)) * (U * (2.0 - (U_42_ * (n / Om))))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 5e-133:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * (((n * math.pow(l_m, 2.0)) * (U * (2.0 - (U_42_ * (n / Om))))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 5e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(n * (l_m ^ 2.0)) * Float64(U * Float64(2.0 - Float64(U_42_ * Float64(n / Om))))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 5e-133)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt((-2.0 * (((n * (l_m ^ 2.0)) * (U * (2.0 - (U_42_ * (n / Om))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * 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$2, 5e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999999e-133

   1. Initial program 25.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in n around 0 51.7%

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

   if 4.9999999999999999e-133 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 67.0%

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

   3. Simplified 70.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 14.4%

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

   5. Taylor expanded in Om around -inf 23.6%

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

   6. Taylor expanded in U around 0 24.1%

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

      1. mul-1-neg 24.1%

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

      2. associate-/l* 24.1%

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

      3. associate-/l* 19.5%

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

      4. associate-/l* 21.7%

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

   8. Simplified 21.7%

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

   9. Taylor expanded in l around inf 43.8%

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

       1. associate-*r* 44.3%

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

       2. *-commutative 44.3%

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

       3. *-commutative 44.3%

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

       4. associate-/l* 46.7%

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

       5. *-commutative 46.7%

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

   11. Simplified 46.7%

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

       1. pow1 46.7%

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

       2. *-commutative 46.7%

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

       3. distribute-lft-out-- 46.7%

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

       4. associate-/l* 44.4%

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

   13. Applied egg-rr 44.4%

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

       1. unpow1 44.4%

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

       2. associate-*r/ 46.7%

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

       3. *-commutative 46.7%

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

       4. associate-*r/ 46.7%

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

   15. Simplified 46.7%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;U \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{{l\_m}^{2} \cdot \left(U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -4.4e-46)
   (sqrt (* (* 2.0 (* n U)) (- t (* 2.0 (/ (pow l_m 2.0) Om)))))
   (if (<= U 2.95e+122)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (* (pow l_m 2.0) (* U* (/ n Om))) Om)))))
     (* (sqrt (* 2.0 U)) (sqrt (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -4.4e-46) {
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (U <= 2.95e+122) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((pow(l_m, 2.0) * (U_42_ * (n / Om))) / Om)))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-4.4d-46)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))
    else if (u <= 2.95d+122) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l_m ** 2.0d0) * (u_42 * (n / om))) / om)))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -4.4e-46) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))));
	} else if (U <= 2.95e+122) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((Math.pow(l_m, 2.0) * (U_42_ * (n / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -4.4e-46:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))
	elif U <= 2.95e+122:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((math.pow(l_m, 2.0) * (U_42_ * (n / Om))) / Om)))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -4.4e-46)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))));
	elseif (U <= 2.95e+122)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64((l_m ^ 2.0) * Float64(U_42_ * Float64(n / Om))) / Om)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -4.4e-46)
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 * ((l_m ^ 2.0) / Om)))));
	elseif (U <= 2.95e+122)
		tmp = sqrt(((2.0 * n) * (U * (t + (((l_m ^ 2.0) * (U_42_ * (n / Om))) / Om)))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -4.4e-46], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 2.95e+122], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -4.4000000000000002e-46

   1. Initial program 64.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in Om around inf 66.1%

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

   if -4.4000000000000002e-46 < U < 2.95000000000000016e122

   1. Initial program 43.1%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in Om around -inf 50.0%

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

   6. Taylor expanded in U around 0 52.0%

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

      1. mul-1-neg 52.0%

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

      2. associate-/l* 52.0%

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

      3. associate-/l* 51.2%

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

      4. associate-/l* 52.8%

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

   8. Simplified 52.8%

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

   9. Taylor expanded in U around 0 49.2%

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

       1. mul-1-neg 49.2%

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

       2. unsub-neg 49.2%

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

       3. associate-*r* 47.4%

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

       4. associate-*r/ 47.9%

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

       5. *-commutative 47.9%

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

   11. Simplified 47.9%

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

   12. Taylor expanded in U* around inf 50.0%

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

       1. mul-1-neg 50.0%

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

       2. associate-*r* 48.8%

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

       3. *-commutative 48.8%

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

       4. associate-/l* 49.2%

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

       5. associate-*r* 49.9%

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

       6. distribute-rgt-neg-in 49.9%

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

       7. distribute-rgt-neg-in 49.9%

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

       8. distribute-neg-frac2 49.9%

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

   14. Simplified 49.9%

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

   if 2.95000000000000016e122 < U

   1. Initial program 59.6%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in t around inf 49.1%

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

      1. pow1/2 52.6%

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

      2. associate-*r* 52.6%

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

      3. unpow-prod-down 65.8%

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

      4. pow1/2 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. unpow1/2 62.2%

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

   9. Simplified 62.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;U \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{{\ell}^{2} \cdot \left(U* \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -1.36 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot {l\_m}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -1.36e+181)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (if (<= t 1.8e+129)
     (sqrt (* (* 2.0 n) (+ (* U t) (* -2.0 (/ (* U (pow l_m 2.0)) Om)))))
     (* (sqrt t) (sqrt (* 2.0 (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -1.36e+181) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else if (t <= 1.8e+129) {
		tmp = sqrt(((2.0 * n) * ((U * t) + (-2.0 * ((U * pow(l_m, 2.0)) / Om)))));
	} else {
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-1.36d+181)) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else if (t <= 1.8d+129) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((-2.0d0) * ((u * (l_m ** 2.0d0)) / om)))))
    else
        tmp = sqrt(t) * sqrt((2.0d0 * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -1.36e+181) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else if (t <= 1.8e+129) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + (-2.0 * ((U * Math.pow(l_m, 2.0)) / Om)))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -1.36e+181:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	elif t <= 1.8e+129:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + (-2.0 * ((U * math.pow(l_m, 2.0)) / Om)))))
	else:
		tmp = math.sqrt(t) * math.sqrt((2.0 * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -1.36e+181)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	elseif (t <= 1.8e+129)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * (l_m ^ 2.0)) / Om)))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.36e+181)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	elseif (t <= 1.8e+129)
		tmp = sqrt(((2.0 * n) * ((U * t) + (-2.0 * ((U * (l_m ^ 2.0)) / Om)))));
	else
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -1.36e+181], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.8e+129], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.35999999999999994e181

   1. Initial program 57.4%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in t around inf 62.3%

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

      1. add-sqr-sqrt 62.3%

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

      2. pow1/2 62.3%

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

      3. pow1/2 66.5%

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

      4. pow-prod-down 32.1%

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

      5. pow2 32.1%

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

      6. associate-*r* 32.1%

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

   7. Applied egg-rr 32.1%

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

      1. unpow1/2 32.1%

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

      2. unpow2 32.1%

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

      3. rem-sqrt-square 67.2%

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

   9. Simplified 67.2%

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

   if -1.35999999999999994e181 < t < 1.8000000000000001e129

   1. Initial program 47.6%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in Om around inf 50.3%

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

   if 1.8000000000000001e129 < t

   1. Initial program 54.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in t around inf 42.8%

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

      1. associate-*r* 46.9%

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

   7. Simplified 46.9%

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

      1. pow1/2 48.9%

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

      2. associate-*l* 44.9%

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

      3. associate-*l* 44.9%

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

      4. metadata-eval 44.9%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. associate-*r* 48.9%

         \[\leadsto {\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      6. unpow-prod-down 62.2%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      7. metadata-eval 62.2%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\color{blue}{0.5}} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      8. metadata-eval 62.2%

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

      9. pow1/2 62.2%

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

   9. Applied egg-rr 62.2%

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

       1. *-commutative 62.2%

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

       2. unpow1/2 62.2%

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

       3. associate-*r* 62.2%

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

       4. *-commutative 62.2%

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

   11. Simplified 62.2%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.36 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -5e+180)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (if (<= t 2.35e+129)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (* (sqrt t) (sqrt (* 2.0 (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -5e+180) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else if (t <= 2.35e+129) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else {
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-5d+180)) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else if (t <= 2.35d+129) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else
        tmp = sqrt(t) * sqrt((2.0d0 * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -5e+180) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else if (t <= 2.35e+129) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -5e+180:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	elif t <= 2.35e+129:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	else:
		tmp = math.sqrt(t) * math.sqrt((2.0 * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -5e+180)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	elseif (t <= 2.35e+129)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -5e+180)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	elseif (t <= 2.35e+129)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	else
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -5e+180], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.35e+129], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.9999999999999996e180

   1. Initial program 57.4%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in t around inf 62.3%

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

      1. add-sqr-sqrt 62.3%

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

      2. pow1/2 62.3%

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

      3. pow1/2 66.5%

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

      4. pow-prod-down 32.1%

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

      5. pow2 32.1%

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

      6. associate-*r* 32.1%

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

   7. Applied egg-rr 32.1%

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

      1. unpow1/2 32.1%

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

      2. unpow2 32.1%

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

      3. rem-sqrt-square 67.2%

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

   9. Simplified 67.2%

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

   if -4.9999999999999996e180 < t < 2.35000000000000004e129

   1. Initial program 47.6%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in n around 0 50.3%

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

   if 2.35000000000000004e129 < t

   1. Initial program 54.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in t around inf 42.8%

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

      1. associate-*r* 46.9%

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

   7. Simplified 46.9%

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

      1. pow1/2 48.9%

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

      2. associate-*l* 44.9%

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

      3. associate-*l* 44.9%

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

      4. metadata-eval 44.9%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. associate-*r* 48.9%

         \[\leadsto {\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      6. unpow-prod-down 62.2%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      7. metadata-eval 62.2%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\color{blue}{0.5}} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      8. metadata-eval 62.2%

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

      9. pow1/2 62.2%

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

   9. Applied egg-rr 62.2%

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

       1. *-commutative 62.2%

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

       2. unpow1/2 62.2%

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

       3. associate-*r* 62.2%

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

       4. *-commutative 62.2%

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

   11. Simplified 62.2%

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

Alternative 9: 40.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot {l\_m}^{2}\right) \cdot \left(U \cdot -4\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -2.85e-300)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (if (<= t 2.65e-163)
     (sqrt (/ (* (* n (pow l_m 2.0)) (* U -4.0)) Om))
     (* (sqrt t) (sqrt (* 2.0 (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -2.85e-300) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else if (t <= 2.65e-163) {
		tmp = sqrt((((n * pow(l_m, 2.0)) * (U * -4.0)) / Om));
	} else {
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-2.85d-300)) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else if (t <= 2.65d-163) then
        tmp = sqrt((((n * (l_m ** 2.0d0)) * (u * (-4.0d0))) / om))
    else
        tmp = sqrt(t) * sqrt((2.0d0 * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -2.85e-300) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else if (t <= 2.65e-163) {
		tmp = Math.sqrt((((n * Math.pow(l_m, 2.0)) * (U * -4.0)) / Om));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -2.85e-300:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	elif t <= 2.65e-163:
		tmp = math.sqrt((((n * math.pow(l_m, 2.0)) * (U * -4.0)) / Om))
	else:
		tmp = math.sqrt(t) * math.sqrt((2.0 * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -2.85e-300)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	elseif (t <= 2.65e-163)
		tmp = sqrt(Float64(Float64(Float64(n * (l_m ^ 2.0)) * Float64(U * -4.0)) / Om));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.85e-300)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	elseif (t <= 2.65e-163)
		tmp = sqrt((((n * (l_m ^ 2.0)) * (U * -4.0)) / Om));
	else
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -2.85e-300], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.65e-163], N[Sqrt[N[(N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.8499999999999999e-300

   1. Initial program 47.2%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in t around inf 42.1%

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

      1. add-sqr-sqrt 42.1%

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

      2. pow1/2 42.1%

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

      3. pow1/2 43.0%

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

      4. pow-prod-down 23.2%

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

      5. pow2 23.2%

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

      6. associate-*r* 23.2%

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

   7. Applied egg-rr 23.2%

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

      1. unpow1/2 23.2%

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

      2. unpow2 23.2%

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

      3. rem-sqrt-square 43.8%

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

   9. Simplified 43.8%

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

   if -2.8499999999999999e-300 < t < 2.65000000000000008e-163

   1. Initial program 36.7%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in Om around inf 45.9%

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

   6. Taylor expanded in l around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. associate-*r* 41.8%

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

      3. *-commutative 41.8%

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

   8. Simplified 41.8%

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

   if 2.65000000000000008e-163 < t

   1. Initial program 55.3%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in t around inf 45.1%

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

      1. associate-*r* 45.2%

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

   7. Simplified 45.2%

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

      1. pow1/2 46.9%

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

      2. associate-*l* 46.0%

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

      3. associate-*l* 46.0%

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

      4. metadata-eval 46.0%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. associate-*r* 46.9%

         \[\leadsto {\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      6. unpow-prod-down 55.1%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      7. metadata-eval 55.1%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\color{blue}{0.5}} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      8. metadata-eval 55.1%

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

      9. pow1/2 55.1%

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

   9. Applied egg-rr 55.1%

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

       1. *-commutative 55.1%

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

       2. unpow1/2 54.2%

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

       3. associate-*r* 54.2%

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

       4. *-commutative 54.2%

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

   11. Simplified 54.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot -4\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;t \leq 1.68 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -3.2e-300)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (if (<= t 1.68e-164)
     (sqrt (* -4.0 (* U (* (pow l_m 2.0) (/ n Om)))))
     (* (sqrt t) (sqrt (* 2.0 (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.2e-300) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else if (t <= 1.68e-164) {
		tmp = sqrt((-4.0 * (U * (pow(l_m, 2.0) * (n / Om)))));
	} else {
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-3.2d-300)) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else if (t <= 1.68d-164) then
        tmp = sqrt(((-4.0d0) * (u * ((l_m ** 2.0d0) * (n / om)))))
    else
        tmp = sqrt(t) * sqrt((2.0d0 * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.2e-300) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else if (t <= 1.68e-164) {
		tmp = Math.sqrt((-4.0 * (U * (Math.pow(l_m, 2.0) * (n / Om)))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -3.2e-300:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	elif t <= 1.68e-164:
		tmp = math.sqrt((-4.0 * (U * (math.pow(l_m, 2.0) * (n / Om)))))
	else:
		tmp = math.sqrt(t) * math.sqrt((2.0 * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -3.2e-300)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	elseif (t <= 1.68e-164)
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64((l_m ^ 2.0) * Float64(n / Om)))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -3.2e-300)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	elseif (t <= 1.68e-164)
		tmp = sqrt((-4.0 * (U * ((l_m ^ 2.0) * (n / Om)))));
	else
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -3.2e-300], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.68e-164], N[Sqrt[N[(-4.0 * N[(U * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.20000000000000021e-300

   1. Initial program 47.2%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in t around inf 42.1%

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

      1. add-sqr-sqrt 42.1%

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

      2. pow1/2 42.1%

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

      3. pow1/2 43.0%

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

      4. pow-prod-down 23.2%

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

      5. pow2 23.2%

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

      6. associate-*r* 23.2%

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

   7. Applied egg-rr 23.2%

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

      1. unpow1/2 23.2%

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

      2. unpow2 23.2%

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

      3. rem-sqrt-square 43.8%

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

   9. Simplified 43.8%

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

   if -3.20000000000000021e-300 < t < 1.67999999999999996e-164

   1. Initial program 36.7%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in Om around -inf 49.6%

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

   6. Taylor expanded in U around 0 49.9%

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

      1. mul-1-neg 49.9%

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

      2. associate-/l* 49.9%

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

      3. associate-/l* 45.6%

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

      4. associate-/l* 45.6%

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

   8. Simplified 45.6%

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

   9. Taylor expanded in l around inf 50.8%

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

       1. associate-*r* 46.7%

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

       2. *-commutative 46.7%

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

       3. *-commutative 46.7%

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

       4. associate-/l* 46.7%

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

       5. *-commutative 46.7%

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

   11. Simplified 46.7%

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

   12. Taylor expanded in n around 0 41.8%

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

       1. associate-/l* 37.6%

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

       2. associate-*r/ 37.6%

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

   14. Simplified 37.6%

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

   if 1.67999999999999996e-164 < t

   1. Initial program 55.3%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in t around inf 45.1%

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

      1. associate-*r* 45.2%

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

   7. Simplified 45.2%

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

      1. pow1/2 46.9%

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

      2. associate-*l* 46.0%

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

      3. associate-*l* 46.0%

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

      4. metadata-eval 46.0%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. associate-*r* 46.9%

         \[\leadsto {\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      6. unpow-prod-down 55.1%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      7. metadata-eval 55.1%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\color{blue}{0.5}} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      8. metadata-eval 55.1%

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

      9. pow1/2 55.1%

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

   9. Applied egg-rr 55.1%

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

       1. *-commutative 55.1%

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

       2. unpow1/2 54.2%

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

       3. associate-*r* 54.2%

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

       4. *-commutative 54.2%

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

   11. Simplified 54.2%

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

Alternative 11: 39.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 7.2e+40)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (* (sqrt (* 2.0 n)) (sqrt (* U t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 7.2e+40) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 7.2d+40) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 7.2e+40) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 7.2e+40:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 7.2e+40)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 7.2e+40)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 7.2e+40], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 7.19999999999999993e40

   1. Initial program 48.1%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around inf 43.3%

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

      1. add-sqr-sqrt 43.3%

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

      2. pow1/2 43.3%

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

      3. pow1/2 43.8%

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

      4. pow-prod-down 25.8%

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

      5. pow2 25.8%

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

      6. associate-*r* 25.8%

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

   7. Applied egg-rr 25.8%

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

      1. unpow1/2 25.8%

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

      2. unpow2 25.8%

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

      3. rem-sqrt-square 44.6%

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

   9. Simplified 44.6%

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

   if 7.19999999999999993e40 < n

   1. Initial program 57.1%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in Om around -inf 40.0%

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

   6. Taylor expanded in U around 0 53.3%

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

      1. mul-1-neg 53.3%

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

      2. associate-/l* 53.3%

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

      3. associate-/l* 54.5%

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

      4. associate-/l* 58.4%

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

   8. Simplified 58.4%

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

      1. sqrt-prod 75.6%

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

      2. +-commutative 75.6%

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

      3. fma-define 75.6%

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

      4. mul-1-neg 75.6%

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

      5. unsub-neg 75.6%

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

      6. associate-*r* 74.5%

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

   10. Applied egg-rr 74.5%

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

       1. *-commutative 74.5%

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

       2. fmm-undef 74.5%

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

       3. associate-*r* 74.5%

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

       4. *-commutative 74.5%

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

       5. associate-*r* 74.4%

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

       6. associate-*r* 69.2%

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

   12. Simplified 69.2%

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

   13. Taylor expanded in t around inf 52.7%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 3.6e-274)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (* (sqrt t) (sqrt (* 2.0 (* n U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 3.6e-274) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else {
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 3.6d-274) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else
        tmp = sqrt(t) * sqrt((2.0d0 * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 3.6e-274) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((2.0 * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 3.6e-274:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	else:
		tmp = math.sqrt(t) * math.sqrt((2.0 * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 3.6e-274)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= 3.6e-274)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	else
		tmp = sqrt(t) * sqrt((2.0 * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 3.6e-274], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.59999999999999983e-274

   1. Initial program 46.3%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 39.9%

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

      1. add-sqr-sqrt 39.9%

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

      2. pow1/2 39.9%

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

      3. pow1/2 40.7%

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

      4. pow-prod-down 22.1%

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

      5. pow2 22.1%

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

      6. associate-*r* 22.1%

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

   7. Applied egg-rr 22.1%

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

      1. unpow1/2 22.1%

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

      2. unpow2 22.1%

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

      3. rem-sqrt-square 41.6%

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

   9. Simplified 41.6%

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

   if 3.59999999999999983e-274 < t

   1. Initial program 53.3%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in t around inf 40.6%

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

      1. associate-*r* 40.7%

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

   7. Simplified 40.7%

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

      1. pow1/2 42.3%

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

      2. associate-*l* 41.4%

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

      3. associate-*l* 41.4%

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

      4. metadata-eval 41.4%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. associate-*r* 42.3%

         \[\leadsto {\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      6. unpow-prod-down 50.6%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      7. metadata-eval 50.6%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\color{blue}{0.5}} \cdot {t}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      8. metadata-eval 50.6%

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

      9. pow1/2 50.6%

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

   9. Applied egg-rr 50.6%

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

       1. *-commutative 50.6%

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

       2. unpow1/2 49.8%

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

       3. associate-*r* 49.8%

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

       4. *-commutative 49.8%

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

   11. Simplified 49.8%

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

Alternative 13: 38.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 3.4 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 3.4e-217)
   (sqrt (fabs (* (* 2.0 U) (* n t))))
   (sqrt (fabs (* 2.0 (* t (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 3.4e-217) {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	} else {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 3.4d-217) then
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    else
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 3.4e-217) {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 3.4e-217:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	else:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 3.4e-217)
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 3.4e-217)
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	else
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 3.4e-217], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.40000000000000016e-217

   1. Initial program 46.4%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in t around inf 45.0%

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

      1. add-sqr-sqrt 45.0%

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

      2. pow1/2 45.0%

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

      3. pow1/2 45.7%

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

      4. pow-prod-down 30.1%

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

      5. pow2 30.1%

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

      6. associate-*r* 30.1%

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

   7. Applied egg-rr 30.1%

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

      1. unpow1/2 30.1%

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

      2. unpow2 30.1%

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

      3. rem-sqrt-square 46.5%

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

   9. Simplified 46.5%

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

   if 3.40000000000000016e-217 < n

   1. Initial program 55.2%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in t around inf 33.1%

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

      1. associate-*r* 38.8%

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

   7. Simplified 38.8%

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

      1. pow1 38.8%

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

      2. associate-*l* 33.1%

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

      3. associate-*l* 33.1%

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

      4. metadata-eval 33.1%

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

      5. metadata-eval 33.1%

         \[\leadsto \sqrt{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\color{blue}{1.5 \cdot 0.3333333333333333} + 0.5\right)}} \]

      6. metadata-eval 33.1%

         \[\leadsto \sqrt{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333 + \color{blue}{1.5 \cdot 0.3333333333333333}\right)}} \]

      7. pow-prod-up 34.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

      8. pow-prod-down 17.5%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

      9. pow2 17.5%

         \[\leadsto \sqrt{{\color{blue}{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{2}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      10. associate-*l* 17.5%

          \[\leadsto \sqrt{{\left({\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{2}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      11. metadata-eval 17.5%

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

   9. Applied egg-rr 17.5%

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

       1. unpow1/2 17.5%

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

       2. unpow2 17.5%

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

       3. rem-sqrt-square 34.7%

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

       4. associate-*r* 40.6%

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

       5. *-commutative 40.6%

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

   11. Simplified 40.6%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.4 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-284}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -3e-284)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (fabs (* 2.0 (* t (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -3e-284) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-3d-284)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -3e-284) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -3e-284:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -3e-284)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= -3e-284)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -3e-284], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3e-284

   1. Initial program 42.8%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in t around inf 43.3%

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

      1. pow1/2 44.1%

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

      2. associate-*l* 44.1%

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

   7. Applied egg-rr 44.1%

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

   if -3e-284 < n

   1. Initial program 57.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 38.6%

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

      1. associate-*r* 42.4%

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

   7. Simplified 42.4%

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

      1. pow1 42.4%

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

      2. associate-*l* 38.6%

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

      3. associate-*l* 38.6%

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

      4. metadata-eval 38.6%

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

      5. metadata-eval 38.6%

         \[\leadsto \sqrt{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\color{blue}{1.5 \cdot 0.3333333333333333} + 0.5\right)}} \]

      6. metadata-eval 38.6%

         \[\leadsto \sqrt{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333 + \color{blue}{1.5 \cdot 0.3333333333333333}\right)}} \]

      7. pow-prod-up 39.4%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

      8. pow-prod-down 20.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

      9. pow2 20.2%

         \[\leadsto \sqrt{{\color{blue}{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{2}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      10. associate-*l* 20.2%

          \[\leadsto \sqrt{{\left({\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{2}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      11. metadata-eval 20.2%

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

   9. Applied egg-rr 20.2%

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

       1. unpow1/2 20.2%

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

       2. unpow2 20.2%

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

       3. rem-sqrt-square 40.1%

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

       4. associate-*r* 44.0%

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

       5. *-commutative 44.0%

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

   11. Simplified 44.0%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-284}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 3 \cdot 10^{+29}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 3e+29)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 3e+29) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = pow((2.0 * (n * (U * t))), 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 (n <= 3d+29) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (n * (u * t))) ** 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 (n <= 3e+29) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 3e+29:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 3e+29)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 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 (n <= 3e+29)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = (2.0 * (n * (U * t))) ^ 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[n, 3e+29], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.9999999999999999e29

   1. Initial program 48.1%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around inf 43.1%

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

      1. associate-*r* 39.7%

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

   7. Simplified 39.7%

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

      1. pow1/2 40.7%

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

      2. associate-*l* 43.6%

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

   9. Applied egg-rr 43.6%

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

   if 2.9999999999999999e29 < n

   1. Initial program 56.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around inf 40.2%

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

      1. pow1/2 42.1%

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

      2. associate-*l* 42.1%

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

   7. Applied egg-rr 42.1%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.3e-225)
   (sqrt (* (* 2.0 n) (* U t)))
   (pow (* 2.0 (* U (* n t))) 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 <= 2.3e-225) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 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 <= 2.3d-225) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 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 <= 2.3e-225) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 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 <= 2.3e-225:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 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 <= 2.3e-225)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 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 <= 2.3e-225)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = (2.0 * (U * (n * t))) ^ 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, 2.3e-225], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2999999999999999e-225

   1. Initial program 59.2%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in t around inf 45.9%

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

   if 2.2999999999999999e-225 < l

   1. Initial program 37.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in t around inf 32.1%

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

      1. associate-*r* 28.5%

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

   7. Simplified 28.5%

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

      1. pow1/2 31.2%

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

      2. associate-*l* 33.9%

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

   9. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 35.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -4.8e-288)
   (sqrt (* (* 2.0 n) (* U t)))
   (sqrt (* 2.0 (* t (* n U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -4.8e-288) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-4.8d-288)) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -4.8e-288) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -4.8e-288:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -4.8e-288)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= -4.8e-288)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -4.8e-288], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.7999999999999997e-288

   1. Initial program 42.8%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in t around inf 43.3%

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

   if -4.7999999999999997e-288 < n

   1. Initial program 57.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 38.6%

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

      1. associate-*r* 42.4%

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

   7. Simplified 42.4%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 8e-217) (sqrt (* 2.0 (* U (* n t)))) (sqrt (* 2.0 (* t (* n U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 8e-217) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 8d-217) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 8e-217) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 8e-217:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 8e-217)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 8e-217)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 8e-217], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.00000000000000066e-217

   1. Initial program 46.4%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in t around inf 45.0%

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

   if 8.00000000000000066e-217 < n

   1. Initial program 55.2%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in t around inf 33.1%

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

      1. associate-*r* 38.8%

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

   7. Simplified 38.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.1% 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 49.8%

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

3. Simplified 52.2%

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

5. Taylor expanded in t around inf 40.3%

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

Reproduce

herbie shell --seed 2024181 
(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*))))))