Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 64.8%

Time: 34.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 64.8% accurate, 0.3× speedup

Math

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

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

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);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * (t + ((U_42_ * (pow(l_m, 2.0) * (n / Om))) / Om))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt((U * (n * (((n * U_42_) / pow(Om, 2.0)) - (2.0 / Om))))) * (l_m * sqrt(2.0));
	}
	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);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + ((U_42_ * (Math.pow(l_m, 2.0) * (n / Om))) / Om))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt((U * (n * (((n * U_42_) / Math.pow(Om, 2.0)) - (2.0 / Om))))) * (l_m * Math.sqrt(2.0));
	}
	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)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * (t + ((U_42_ * (math.pow(l_m, 2.0) * (n / Om))) / Om))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt((U * (n * (((n * U_42_) / math.pow(Om, 2.0)) - (2.0 / Om))))) * (l_m * math.sqrt(2.0))
	return tmp

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.1%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in Om around inf 16.5%

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

      1. *-commutative 16.5%

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

      2. associate-/l* 16.4%

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

      3. distribute-lft-out 16.4%

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

   7. Simplified 16.4%

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

   8. Taylor expanded in U around 0 39.7%

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

      1. mul-1-neg 39.7%

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

   10. Simplified 39.7%

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

   11. Taylor expanded in U* around inf 32.4%

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

       1. mul-1-neg 32.4%

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

       2. associate-/l* 32.4%

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

       3. associate-/l* 39.9%

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

   13. Simplified 39.9%

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

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

   1. Initial program 69.3%

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

   3. Simplified 73.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 10.5%

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

   5. Taylor expanded in Om around inf 15.4%

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

      1. *-commutative 15.4%

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

      2. associate-/l* 15.4%

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

      3. distribute-lft-out 41.8%

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

   7. Simplified 41.8%

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

   8. Taylor expanded in U around 0 43.2%

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

      1. mul-1-neg 43.2%

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

   10. Simplified 43.2%

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

   11. Taylor expanded in l around inf 40.2%

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

       1. associate-*r/ 40.2%

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

       2. metadata-eval 40.2%

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

   13. Simplified 40.2%

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

4. Final simplification 63.4%

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

Alternative 2: 60.5% accurate, 0.5× speedup

Math

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

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

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * (t - ((pow(l_m, 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (n * l_m) * (sqrt(2.0) * (sqrt((U * U_42_)) / 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);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - ((Math.pow(l_m, 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (n * l_m) * (Math.sqrt(2.0) * (Math.sqrt((U * U_42_)) / 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)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * (t - ((math.pow(l_m, 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = (n * l_m) * (math.sqrt(2.0) * (math.sqrt((U * U_42_)) / Om))
	return tmp

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.4%

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

   3. Simplified 17.1%

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

   5. Taylor expanded in Om around inf 21.5%

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

      1. *-commutative 21.5%

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

      2. associate-/l* 21.5%

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

      3. distribute-lft-out 21.6%

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

   7. Simplified 21.6%

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

   8. Taylor expanded in U around 0 39.9%

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

      1. mul-1-neg 39.9%

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

   10. Simplified 39.9%

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

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

   1. Initial program 69.3%

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

   3. Simplified 73.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in U* around inf 48.1%

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

      1. associate-*l/ 48.1%

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

   7. Simplified 48.1%

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

      1. associate-/l* 48.2%

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

      2. associate-*r* 48.2%

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

      3. *-commutative 48.2%

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

   9. Applied egg-rr 48.2%

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

       1. associate-*l* 48.2%

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

   11. Simplified 48.2%

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

4. Final simplification 64.4%

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

Alternative 3: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{U* \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{{l\_m}^{2} \cdot \left(2 - \frac{n \cdot U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{l\_m}^{2} \cdot \left(n \cdot \frac{U - U*}{Om}\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -1.85e+118)
   (sqrt (* 2.0 (* U (* n (+ t (/ (* U* (* (pow l_m 2.0) (/ n Om))) Om))))))
   (if (<= n 1.35e+24)
     (sqrt
      (*
       2.0
       (* U (* n (- t (/ (* (pow l_m 2.0) (- 2.0 (/ (* n U*) Om))) Om))))))
     (sqrt
      (*
       (* 2.0 (* n U))
       (- t (/ (* (pow l_m 2.0) (* n (/ (- U U*) Om))) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -1.85e+118) {
		tmp = sqrt((2.0 * (U * (n * (t + ((U_42_ * (pow(l_m, 2.0) * (n / Om))) / Om))))));
	} else if (n <= 1.35e+24) {
		tmp = sqrt((2.0 * (U * (n * (t - ((pow(l_m, 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((pow(l_m, 2.0) * (n * ((U - U_42_) / Om))) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-1.85d+118)) then
        tmp = sqrt((2.0d0 * (u * (n * (t + ((u_42 * ((l_m ** 2.0d0) * (n / om))) / om))))))
    else if (n <= 1.35d+24) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (((l_m ** 2.0d0) * (2.0d0 - ((n * u_42) / om))) / om))))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (((l_m ** 2.0d0) * (n * ((u - u_42) / om))) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -1.85e+118) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + ((U_42_ * (Math.pow(l_m, 2.0) * (n / Om))) / Om))))));
	} else if (n <= 1.35e+24) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - ((Math.pow(l_m, 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((Math.pow(l_m, 2.0) * (n * ((U - U_42_) / Om))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -1.85e+118:
		tmp = math.sqrt((2.0 * (U * (n * (t + ((U_42_ * (math.pow(l_m, 2.0) * (n / Om))) / Om))))))
	elif n <= 1.35e+24:
		tmp = math.sqrt((2.0 * (U * (n * (t - ((math.pow(l_m, 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((math.pow(l_m, 2.0) * (n * ((U - U_42_) / Om))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -1.85e+118)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(U_42_ * Float64((l_m ^ 2.0) * Float64(n / Om))) / Om))))));
	elseif (n <= 1.35e+24)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(Float64((l_m ^ 2.0) * Float64(2.0 - Float64(Float64(n * U_42_) / Om))) / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64((l_m ^ 2.0) * Float64(n * Float64(Float64(U - U_42_) / Om))) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= -1.85e+118)
		tmp = sqrt((2.0 * (U * (n * (t + ((U_42_ * ((l_m ^ 2.0) * (n / Om))) / Om))))));
	elseif (n <= 1.35e+24)
		tmp = sqrt((2.0 * (U * (n * (t - (((l_m ^ 2.0) * (2.0 - ((n * U_42_) / Om))) / Om))))));
	else
		tmp = sqrt(((2.0 * (n * U)) * (t - (((l_m ^ 2.0) * (n * ((U - U_42_) / Om))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -1.85e+118], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.35e+24], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.84999999999999993e118

   1. Initial program 51.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in Om around inf 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-/l* 43.7%

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

      3. distribute-lft-out 54.4%

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

   7. Simplified 54.4%

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

   8. Taylor expanded in U around 0 61.7%

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

      1. mul-1-neg 61.7%

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

   10. Simplified 61.7%

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

   11. Taylor expanded in U* around inf 76.5%

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

       1. mul-1-neg 76.5%

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

       2. associate-/l* 79.8%

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

       3. associate-/l* 79.8%

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

   13. Simplified 79.8%

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

   if -1.84999999999999993e118 < n < 1.35e24

   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 52.3%

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

   5. Taylor expanded in Om around inf 48.2%

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

      1. *-commutative 48.2%

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

      2. associate-/l* 48.7%

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

      3. distribute-lft-out 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in U around 0 54.1%

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

      1. mul-1-neg 54.1%

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

   10. Simplified 54.1%

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

   if 1.35e24 < n

   1. Initial program 58.0%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in Om around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 44.0%

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

      3. distribute-lft-out 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in n around inf 57.4%

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

      1. div-sub 59.2%

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

   10. Simplified 59.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 - \frac{n \cdot U*}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \frac{U - U*}{Om}\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(n \cdot l\_m\right)}^{2}\\ \mathbf{if}\;Om \leq -9.8 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;Om \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot t\_1\right)} \cdot \frac{-1}{Om}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(\left(U \cdot U*\right) \cdot t\_1\right)\right)}^{0.5}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (* n l_m) 2.0)))
   (if (<= Om -9.8e-113)
     (sqrt (* (* 2.0 (* n U)) (- t (/ (* 2.0 (pow l_m 2.0)) Om))))
     (if (<= Om -5e-310)
       (* (sqrt (* (* 2.0 U) (* U* t_1))) (/ -1.0 Om))
       (if (<= Om 1.5e-138)
         (/ (pow (* 2.0 (* (* U U*) t_1)) 0.5) Om)
         (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l_m 2.0) Om))))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow((n * l_m), 2.0);
	double tmp;
	if (Om <= -9.8e-113) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * pow(l_m, 2.0)) / Om))));
	} else if (Om <= -5e-310) {
		tmp = sqrt(((2.0 * U) * (U_42_ * t_1))) * (-1.0 / Om);
	} else if (Om <= 1.5e-138) {
		tmp = pow((2.0 * ((U * U_42_) * t_1)), 0.5) / Om;
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l_m, 2.0) / Om)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (n * l_m) ** 2.0d0
    if (om <= (-9.8d-113)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((2.0d0 * (l_m ** 2.0d0)) / om))))
    else if (om <= (-5d-310)) then
        tmp = sqrt(((2.0d0 * u) * (u_42 * t_1))) * ((-1.0d0) / om)
    else if (om <= 1.5d-138) then
        tmp = ((2.0d0 * ((u * u_42) * t_1)) ** 0.5d0) / om
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.pow((n * l_m), 2.0);
	double tmp;
	if (Om <= -9.8e-113) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om))));
	} else if (Om <= -5e-310) {
		tmp = Math.sqrt(((2.0 * U) * (U_42_ * t_1))) * (-1.0 / Om);
	} else if (Om <= 1.5e-138) {
		tmp = Math.pow((2.0 * ((U * U_42_) * t_1)), 0.5) / Om;
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.pow((n * l_m), 2.0)
	tmp = 0
	if Om <= -9.8e-113:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om))))
	elif Om <= -5e-310:
		tmp = math.sqrt(((2.0 * U) * (U_42_ * t_1))) * (-1.0 / Om)
	elif Om <= 1.5e-138:
		tmp = math.pow((2.0 * ((U * U_42_) * t_1)), 0.5) / Om
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * l_m) ^ 2.0
	tmp = 0.0
	if (Om <= -9.8e-113)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om))));
	elseif (Om <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(2.0 * U) * Float64(U_42_ * t_1))) * Float64(-1.0 / Om));
	elseif (Om <= 1.5e-138)
		tmp = Float64((Float64(2.0 * Float64(Float64(U * U_42_) * t_1)) ^ 0.5) / Om);
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * l_m) ^ 2.0;
	tmp = 0.0;
	if (Om <= -9.8e-113)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l_m ^ 2.0)) / Om))));
	elseif (Om <= -5e-310)
		tmp = sqrt(((2.0 * U) * (U_42_ * t_1))) * (-1.0 / Om);
	elseif (Om <= 1.5e-138)
		tmp = ((2.0 * ((U * U_42_) * t_1)) ^ 0.5) / Om;
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l_m ^ 2.0) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(n * l$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[Om, -9.8e-113], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -5e-310], N[(N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(U$42$ * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 1.5e-138], N[(N[Power[N[(2.0 * N[(N[(U * U$42$), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / Om), $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -9.8000000000000006e-113

   1. Initial program 56.4%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in Om around inf 52.9%

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

      1. associate-*r/ 52.9%

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

   7. Simplified 52.9%

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

   if -9.8000000000000006e-113 < Om < -4.999999999999985e-310

   1. Initial program 38.3%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in U* around inf 28.2%

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

      1. associate-*r/ 28.2%

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

   7. Simplified 28.2%

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

      1. pow1/2 28.2%

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

      2. div-inv 28.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)\right) \cdot \frac{1}{{Om}^{2}}\right)}}^{0.5} \]

      3. unpow-prod-down 28.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)\right)}^{0.5} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5}} \]

      4. pow1/2 28.2%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5} \]

      5. associate-*r* 28.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5} \]

      6. pow-prod-down 38.1%

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \color{blue}{{\left(\ell \cdot n\right)}^{2}}\right)} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5} \]

      7. pow-flip 38.1%

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

      8. metadata-eval 38.1%

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

   9. Applied egg-rr 38.1%

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

       1. unpow1/2 38.1%

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

   11. Simplified 38.1%

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

   12. Taylor expanded in Om around -inf 57.7%

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

   if -4.999999999999985e-310 < Om < 1.5e-138

   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 38.0%

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

   5. Taylor expanded in U* around inf 40.2%

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

      1. associate-*r/ 40.2%

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

   7. Simplified 40.2%

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

      1. pow1/2 40.6%

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

      2. div-inv 40.6%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)\right) \cdot \frac{1}{{Om}^{2}}\right)}}^{0.5} \]

      3. unpow-prod-down 40.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)\right)}^{0.5} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5}} \]

      4. pow1/2 40.2%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5} \]

      5. associate-*r* 40.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5} \]

      6. pow-prod-down 44.9%

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \color{blue}{{\left(\ell \cdot n\right)}^{2}}\right)} \cdot {\left(\frac{1}{{Om}^{2}}\right)}^{0.5} \]

      7. pow-flip 44.9%

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

      8. metadata-eval 44.9%

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

   9. Applied egg-rr 44.9%

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

       1. unpow1/2 44.9%

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

   11. Simplified 44.9%

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

       1. pow1 44.9%

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

       2. *-commutative 44.9%

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

       3. sqrt-pow1 55.4%

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

       4. metadata-eval 55.4%

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

       5. inv-pow 55.4%

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

       6. associate-*l* 55.4%

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

       7. *-commutative 55.4%

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

   13. Applied egg-rr 55.4%

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

       1. unpow1 55.4%

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

       2. associate-*l/ 55.3%

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

       3. *-lft-identity 55.3%

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

   15. Simplified 55.3%

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

       1. pow1/2 55.3%

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

       2. associate-*r* 58.9%

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

   17. Applied egg-rr 58.9%

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

   if 1.5e-138 < Om

   1. Initial program 51.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in n around 0 54.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.8 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;Om \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot {\left(n \cdot \ell\right)}^{2}\right)} \cdot \frac{-1}{Om}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(\left(U \cdot U*\right) \cdot {\left(n \cdot \ell\right)}^{2}\right)\right)}^{0.5}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{+19}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{U* \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{l\_m}^{2} \cdot \left(n \cdot \frac{U - U*}{Om}\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 1e+19)
   (sqrt (* 2.0 (* U (* n (+ t (/ (* U* (* (pow l_m 2.0) (/ n Om))) Om))))))
   (sqrt
    (* (* 2.0 (* n U)) (- t (/ (* (pow l_m 2.0) (* n (/ (- U U*) Om))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 1e+19) {
		tmp = sqrt((2.0 * (U * (n * (t + ((U_42_ * (pow(l_m, 2.0) * (n / Om))) / Om))))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((pow(l_m, 2.0) * (n * ((U - U_42_) / Om))) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 1d+19) then
        tmp = sqrt((2.0d0 * (u * (n * (t + ((u_42 * ((l_m ** 2.0d0) * (n / om))) / om))))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (((l_m ** 2.0d0) * (n * ((u - u_42) / om))) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 1e+19) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + ((U_42_ * (Math.pow(l_m, 2.0) * (n / Om))) / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((Math.pow(l_m, 2.0) * (n * ((U - U_42_) / Om))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 1e+19:
		tmp = math.sqrt((2.0 * (U * (n * (t + ((U_42_ * (math.pow(l_m, 2.0) * (n / Om))) / Om))))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((math.pow(l_m, 2.0) * (n * ((U - U_42_) / Om))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 1e+19)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(U_42_ * Float64((l_m ^ 2.0) * Float64(n / Om))) / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64((l_m ^ 2.0) * Float64(n * Float64(Float64(U - U_42_) / Om))) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 1e+19)
		tmp = sqrt((2.0 * (U * (n * (t + ((U_42_ * ((l_m ^ 2.0) * (n / Om))) / Om))))));
	else
		tmp = sqrt(((2.0 * (n * U)) * (t - (((l_m ^ 2.0) * (n * ((U - U_42_) / Om))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 1e+19], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1e19

   1. Initial program 48.6%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in Om around inf 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-/l* 48.0%

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

      3. distribute-lft-out 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in U around 0 55.1%

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

      1. mul-1-neg 55.1%

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

   10. Simplified 55.1%

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

   11. Taylor expanded in U* around inf 51.2%

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

       1. mul-1-neg 51.2%

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

       2. associate-/l* 52.1%

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

       3. associate-/l* 54.0%

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

   13. Simplified 54.0%

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

   if 1e19 < n

   1. Initial program 58.0%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in Om around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 44.0%

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

      3. distribute-lft-out 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in n around inf 57.4%

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

      1. div-sub 59.2%

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

   10. Simplified 59.2%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{+19}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \frac{U - U*}{Om}\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.5 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\left(\sqrt{U \cdot U*} \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \frac{1}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.5e+155)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l_m 2.0) Om)))))))
   (* l_m (* (* (sqrt (* U U*)) (* n (sqrt 2.0))) (/ 1.0 Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.5e+155) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = l_m * ((sqrt((U * U_42_)) * (n * sqrt(2.0))) * (1.0 / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.5d+155) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))))
    else
        tmp = l_m * ((sqrt((u * u_42)) * (n * sqrt(2.0d0))) * (1.0d0 / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.5e+155) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = l_m * ((Math.sqrt((U * U_42_)) * (n * Math.sqrt(2.0))) * (1.0 / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.5e+155:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))))
	else:
		tmp = l_m * ((math.sqrt((U * U_42_)) * (n * math.sqrt(2.0))) * (1.0 / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.5e+155)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))))));
	else
		tmp = Float64(l_m * Float64(Float64(sqrt(Float64(U * U_42_)) * Float64(n * sqrt(2.0))) * Float64(1.0 / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.5e+155)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l_m ^ 2.0) / Om)))))));
	else
		tmp = l_m * ((sqrt((U * U_42_)) * (n * sqrt(2.0))) * (1.0 / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.5e+155], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5000000000000001e155

   1. Initial program 56.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in n around 0 50.4%

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

   if 1.5000000000000001e155 < l

   1. Initial program 12.8%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in U* around inf 39.6%

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

      1. associate-*l/ 39.7%

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

   7. Simplified 39.7%

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

      1. div-inv 39.7%

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

      2. associate-*l* 39.8%

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

   9. Applied egg-rr 39.8%

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

       1. associate-*l* 42.4%

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

   11. Simplified 42.4%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(\sqrt{U \cdot U*} \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \frac{1}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.9 \cdot 10^{+156}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\left(\sqrt{U \cdot U*} \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \frac{1}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.9e+156)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (* l_m (* (* (sqrt (* U U*)) (* n (sqrt 2.0))) (/ 1.0 Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.9e+156) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = l_m * ((sqrt((U * U_42_)) * (n * sqrt(2.0))) * (1.0 / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.9d+156) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = l_m * ((sqrt((u * u_42)) * (n * sqrt(2.0d0))) * (1.0d0 / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.9e+156) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = l_m * ((Math.sqrt((U * U_42_)) * (n * Math.sqrt(2.0))) * (1.0 / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.9e+156:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = l_m * ((math.sqrt((U * U_42_)) * (n * math.sqrt(2.0))) * (1.0 / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.9e+156)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(l_m * Float64(Float64(sqrt(Float64(U * U_42_)) * Float64(n * sqrt(2.0))) * Float64(1.0 / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.9e+156)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = l_m * ((sqrt((U * U_42_)) * (n * sqrt(2.0))) * (1.0 / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.9e+156], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(l$95$m * N[(N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.90000000000000012e156

   1. Initial program 56.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in t around inf 44.4%

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

      1. pow1/2 47.7%

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

   7. Applied egg-rr 47.7%

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

   if 1.90000000000000012e156 < l

   1. Initial program 12.8%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in U* around inf 39.6%

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

      1. associate-*l/ 39.7%

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

   7. Simplified 39.7%

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

      1. div-inv 39.7%

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

      2. associate-*l* 39.8%

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

   9. Applied egg-rr 39.8%

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

       1. associate-*l* 42.4%

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

   11. Simplified 42.4%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{+156}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(\sqrt{U \cdot U*} \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \frac{1}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.2% accurate, 1.0× speedup

Math

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

FPCore

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

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 + ((U_42_ * (pow(l_m, 2.0) * (n / Om))) / Om))))));
}

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 + ((u_42 * ((l_m ** 2.0d0) * (n / om))) / om))))))
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 + ((U_42_ * (Math.pow(l_m, 2.0) * (n / Om))) / Om))))));
}

Python

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

Julia

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

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * (t + ((U_42_ * ((l_m ^ 2.0) * (n / Om))) / Om))))));
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 * N[(t + N[(N[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Simplified 55.0%

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

5. Taylor expanded in Om around inf 45.3%

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

   1. *-commutative 45.3%

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

   2. associate-/l* 47.2%

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

   3. distribute-lft-out 51.4%

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

7. Simplified 51.4%

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

8. Taylor expanded in U around 0 53.3%

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

   1. mul-1-neg 53.3%

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

10. Simplified 53.3%

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

11. Taylor expanded in U* around inf 49.8%

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

    1. mul-1-neg 49.8%

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

    2. associate-/l* 51.7%

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

    3. associate-/l* 53.1%

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

13. Simplified 53.1%

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

14. Final simplification 53.1%

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

Alternative 9: 40.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.55 \cdot 10^{+159}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.55e+159)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (* (sqrt (* U U*)) (* l_m (/ (* n (sqrt 2.0)) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.55e+159) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((U * U_42_)) * (l_m * ((n * sqrt(2.0)) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.55d+159) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((u * u_42)) * (l_m * ((n * sqrt(2.0d0)) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.55e+159) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((U * U_42_)) * (l_m * ((n * Math.sqrt(2.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.55e+159:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((U * U_42_)) * (l_m * ((n * math.sqrt(2.0)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.55e+159)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(Float64(n * sqrt(2.0)) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.55e+159)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((U * U_42_)) * (l_m * ((n * sqrt(2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.55e+159], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5499999999999999e159

   1. Initial program 56.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in t around inf 44.4%

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

      1. pow1/2 47.7%

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

   7. Applied egg-rr 47.7%

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

   if 1.5499999999999999e159 < l

   1. Initial program 12.8%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in U* around inf 39.6%

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

      1. associate-/l* 42.3%

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

   7. Simplified 42.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+159}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-193}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 8e-193)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (* (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 <= 8e-193) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} 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 <= 8d-193) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    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 <= 8e-193) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} 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 <= 8e-193:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	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 <= 8e-193)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	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 <= 8e-193)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	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, 8e-193], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $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 < 8.0000000000000004e-193

   1. Initial program 48.9%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in t around inf 43.0%

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

      1. pow1/2 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 8.0000000000000004e-193 < n

   1. Initial program 53.5%

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

   3. Simplified 60.1%

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

      1. sqrt-prod 68.2%

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

      2. fma-undefine 68.2%

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

      3. associate-*r* 68.5%

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

      4. +-commutative 68.5%

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

      5. *-commutative 68.5%

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

      6. fma-define 68.5%

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

      7. associate-*r/ 62.1%

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

      8. associate-*r/ 62.1%

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

      9. pow2 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. *-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in t around inf 40.1%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-193}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.4% accurate, 2.1× speedup

Math

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

FPCore

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

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (2.0d0 * (u * (n * t))) ** 0.5d0
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.pow((2.0 * (U * (n * t))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

3. Simplified 55.4%

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

5. Taylor expanded in t around inf 38.9%

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

   1. pow1/2 42.4%

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

7. Applied egg-rr 42.4%

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

8. Final simplification 42.4%

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

Alternative 12: 36.6% 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 50.6%

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

3. Simplified 55.4%

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

5. Taylor expanded in t around inf 38.9%

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

6. Final simplification 38.9%

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

Alternative 13: 36.5% accurate, 2.1× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Simplified 55.4%

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

5. Taylor expanded in t around inf 38.9%

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

   1. pow1/2 42.4%

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

7. Applied egg-rr 42.4%

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

   1. unpow1/2 38.9%

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

   2. associate-*r* 38.9%

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

9. Applied egg-rr 38.9%

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

10. Final simplification 38.9%

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

Reproduce

herbie shell --seed 2024053 
(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*))))))