Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 64.0%

Time: 30.1s

Alternatives: 19

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: 50.5% 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: 64.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt[3]{{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)\right)}^{0.75}}\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot t} \cdot \sqrt{U}\right) - \frac{{l_m}^{2} \cdot \sqrt{2}}{Om} \cdot \sqrt{\frac{n \cdot U}{t}}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \left(t_2 \cdot t_2\right)\\ \end{array} \end{array} \]

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)) (- U* U)))
        (t_2
         (cbrt
          (pow
           (* U (* n (fma (* n (pow Om -2.0)) (- U* U) (/ -2.0 Om))))
           0.75)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_4 0.0)
     (-
      (* (sqrt 2.0) (* (sqrt (* n t)) (sqrt U)))
      (* (/ (* (pow l_m 2.0) (sqrt 2.0)) Om) (sqrt (/ (* n U) t))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (* (* l_m (sqrt 2.0)) (* t_2 t_2))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = cbrt(pow((U * (n * fma((n * pow(Om, -2.0)), (U_42_ - U), (-2.0 / Om)))), 0.75));
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = (sqrt(2.0) * (sqrt((n * t)) * sqrt(U))) - (((pow(l_m, 2.0) * sqrt(2.0)) / Om) * sqrt(((n * U) / t)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * sqrt(2.0)) * (t_2 * t_2);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = cbrt((Float64(U * Float64(n * fma(Float64(n * (Om ^ -2.0)), Float64(U_42_ - U), Float64(-2.0 / Om)))) ^ 0.75))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(Float64(n * t)) * sqrt(U))) - Float64(Float64(Float64((l_m ^ 2.0) * sqrt(2.0)) / Om) * sqrt(Float64(Float64(n * U) / t))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * Float64(t_2 * t_2));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[N[(U * N[(n * N[(N[(n * N[Power[Om, -2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.75], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$2), $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 11.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in Om around inf 32.2%

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

      1. sqrt-prod 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. *-commutative 41.0%

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

   9. Simplified 41.0%

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

   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 69.2%

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

      1. associate-*l/ 74.1%

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

   4. Applied egg-rr 74.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in l around inf 29.3%

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

      1. associate-/l* 29.2%

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

      2. associate-*r/ 29.2%

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

      3. metadata-eval 29.2%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in l around 0 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-*r* 24.1%

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

      3. associate-*l/ 24.0%

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

      4. fma-neg 24.0%

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

      5. associate-*r/ 24.0%

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

      6. metadata-eval 24.0%

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

      7. distribute-neg-frac 24.0%

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

      8. metadata-eval 24.0%

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

   10. Simplified 24.0%

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

       1. add-cbrt-cube 24.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)} \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)}\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)}}} \]

       2. pow1/3 23.1%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{{\left(\left(\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)} \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)}\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)}\right)}^{0.3333333333333333}} \]

   12. Applied egg-rr 23.2%

       \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{{\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   13. Step-by-step derivation

       1. unpow1/3 24.1%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)}^{1.5}}} \]

       2. *-commutative 24.1%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(\color{blue}{\left(U \cdot n\right)} \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)}^{1.5}} \]

       3. associate-*l* 29.8%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\color{blue}{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)\right)}}^{1.5}} \]

       4. fma-def 29.8%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(U \cdot \left(n \cdot \color{blue}{\left(\left(n \cdot {Om}^{-2}\right) \cdot \left(U* - U\right) + \frac{-2}{Om}\right)}\right)\right)}^{1.5}} \]

       5. associate-*l* 23.8%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(U \cdot \left(n \cdot \left(\color{blue}{n \cdot \left({Om}^{-2} \cdot \left(U* - U\right)\right)} + \frac{-2}{Om}\right)\right)\right)}^{1.5}} \]

       6. fma-def 23.8%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \sqrt[3]{{\left(U \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)}\right)\right)}^{1.5}} \]

   14. Simplified 23.8%

       \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt[3]{{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)\right)}^{1.5}}} \]
   15. Step-by-step derivation

       1. pow1/3 22.9%

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

       2. sqr-pow 22.9%

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

       3. unpow-prod-down 25.1%

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

       4. associate-*r* 21.2%

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

       5. *-commutative 21.2%

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

       6. metadata-eval 21.2%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left({\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{\color{blue}{0.75}}\right)}^{0.3333333333333333} \cdot {\left({\left(U \cdot \left(n \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)\right)}^{\left(\frac{1.5}{2}\right)}\right)}^{0.3333333333333333}\right) \]

       7. associate-*r* 21.2%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left({\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\color{blue}{\left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}}^{\left(\frac{1.5}{2}\right)}\right)}^{0.3333333333333333}\right) \]

       8. *-commutative 21.2%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left({\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\left(\color{blue}{\left(n \cdot U\right)} \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{\left(\frac{1.5}{2}\right)}\right)}^{0.3333333333333333}\right) \]

       9. metadata-eval 21.2%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left({\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{\color{blue}{0.75}}\right)}^{0.3333333333333333}\right) \]

   16. Applied egg-rr 21.2%

       \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\left({\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}\right)}^{0.3333333333333333}\right)} \]
   17. Step-by-step derivation

       1. unpow1/3 21.4%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}} \cdot {\left({\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}\right)}^{0.3333333333333333}\right) \]

       2. unpow1/3 22.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}} \cdot \color{blue}{\sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}}\right) \]

       3. *-commutative 22.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\left(\color{blue}{\left(U \cdot n\right)} \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}} \cdot \sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}\right) \]

       4. associate-*l* 22.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\color{blue}{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)\right)}}^{0.75}} \cdot \sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}\right) \]

       5. fma-def 22.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\left(U \cdot \left(n \cdot \color{blue}{\left(n \cdot \left({Om}^{-2} \cdot \left(U* - U\right)\right) + \frac{-2}{Om}\right)}\right)\right)}^{0.75}} \cdot \sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}\right) \]

       6. associate-*r* 22.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\left(U \cdot \left(n \cdot \left(\color{blue}{\left(n \cdot {Om}^{-2}\right) \cdot \left(U* - U\right)} + \frac{-2}{Om}\right)\right)\right)}^{0.75}} \cdot \sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}\right) \]

       7. fma-def 22.0%

          \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\left(U \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)}\right)\right)}^{0.75}} \cdot \sqrt[3]{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {Om}^{-2} \cdot \left(U* - U\right), \frac{-2}{Om}\right)\right)}^{0.75}}\right) \]

   18. Simplified 32.0%

       \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt[3]{{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)\right)}^{0.75}} \cdot \sqrt[3]{{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)\right)}^{0.75}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot t} \cdot \sqrt{U}\right) - \frac{{\ell}^{2} \cdot \sqrt{2}}{Om} \cdot \sqrt{\frac{n \cdot U}{t}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(\sqrt[3]{{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)\right)}^{0.75}} \cdot \sqrt[3]{{\left(U \cdot \left(n \cdot \mathsf{fma}\left(n \cdot {Om}^{-2}, U* - U, \frac{-2}{Om}\right)\right)\right)}^{0.75}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot t} \cdot \sqrt{U}\right) - \frac{{l_m}^{2} \cdot \sqrt{2}}{Om} \cdot \sqrt{\frac{n \cdot U}{t}}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 0.0)
     (-
      (* (sqrt 2.0) (* (sqrt (* n t)) (sqrt U)))
      (* (/ (* (pow l_m 2.0) (sqrt 2.0)) Om) (sqrt (/ (* n U) t))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (* l_m (sqrt 2.0))
        (sqrt
         (*
          U
          (* n (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 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 t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (sqrt(2.0) * (sqrt((n * t)) * sqrt(U))) - (((pow(l_m, 2.0) * sqrt(2.0)) / Om) * sqrt(((n * U) / t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (Math.sqrt(2.0) * (Math.sqrt((n * t)) * Math.sqrt(U))) - (((Math.pow(l_m, 2.0) * Math.sqrt(2.0)) / Om) * Math.sqrt(((n * U) / t)));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * (((n * (U_42_ - U)) / Math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_3 <= 0.0:
		tmp = (math.sqrt(2.0) * (math.sqrt((n * t)) * math.sqrt(U))) - (((math.pow(l_m, 2.0) * math.sqrt(2.0)) / Om) * math.sqrt(((n * U) / t)))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * (((n * (U_42_ - U)) / math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(Float64(n * t)) * sqrt(U))) - Float64(Float64(Float64((l_m ^ 2.0) * sqrt(2.0)) / Om) * sqrt(Float64(Float64(n * U) / t))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(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_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = (sqrt(2.0) * (sqrt((n * t)) * sqrt(U))) - ((((l_m ^ 2.0) * sqrt(2.0)) / Om) * sqrt(((n * U) / t)));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * (((n * (U_42_ - U)) / (Om ^ 2.0)) + (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$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 11.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in Om around inf 32.2%

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

      1. sqrt-prod 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. *-commutative 41.0%

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

   9. Simplified 41.0%

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

   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 69.2%

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

      1. associate-*l/ 74.1%

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

   4. Applied egg-rr 74.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in l around inf 31.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot t} \cdot \sqrt{U}\right) - \frac{{\ell}^{2} \cdot \sqrt{2}}{Om} \cdot \sqrt{\frac{n \cdot U}{t}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((n * t)) * sqrt((2.0 * U));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = (-4.0 * ((U * (n * (l_m ^ 2.0))) / Om)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.9%

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

   3. Simplified 13.7%

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

   5. Taylor expanded in l around 0 30.5%

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

      1. pow1/2 30.5%

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

      2. associate-*r* 30.5%

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

      3. unpow-prod-down 44.8%

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

      4. pow1/2 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1/2 44.8%

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

   9. Simplified 44.8%

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

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

   1. Initial program 69.2%

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

      1. associate-*l/ 74.1%

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

   4. Applied egg-rr 74.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in n around 0 7.2%

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

      1. pow1/2 32.6%

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

      2. associate-*r* 32.6%

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

      3. +-commutative 32.6%

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

      4. unpow2 32.6%

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

      5. associate-*l/ 34.6%

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

      6. *-commutative 34.6%

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

      7. fma-def 34.6%

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

      8. associate-*l/ 32.6%

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

      9. unpow2 32.6%

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

   7. Applied egg-rr 32.6%

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

   8. Taylor expanded in l around inf 41.8%

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

4. Final simplification 64.4%

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

Alternative 4: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 0.0)
     (* (sqrt (* n t)) (sqrt (* 2.0 U)))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (* l_m (sqrt 2.0))
        (sqrt
         (*
          U
          (* n (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 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 t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * t)) * sqrt((2.0 * U));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((n * t)) * Math.sqrt((2.0 * U));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * (((n * (U_42_ - U)) / Math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((n * t)) * math.sqrt((2.0 * U))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * (((n * (U_42_ - U)) / math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(n * t)) * sqrt(Float64(2.0 * U)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(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_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((n * t)) * sqrt((2.0 * U));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * (((n * (U_42_ - U)) / (Om ^ 2.0)) + (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$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 11.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in l around 0 35.2%

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

      1. pow1/2 35.2%

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

      2. associate-*r* 35.2%

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

      3. unpow-prod-down 40.9%

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

      4. pow1/2 40.9%

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

   7. Applied egg-rr 40.9%

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

      1. unpow1/2 40.9%

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

   9. Simplified 40.9%

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

   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 69.2%

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

      1. associate-*l/ 74.1%

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

   4. Applied egg-rr 74.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in l around inf 31.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 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]), $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 11.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in l around 0 35.2%

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

      1. pow1/2 35.2%

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

      2. associate-*r* 35.2%

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

      3. unpow-prod-down 40.9%

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

      4. pow1/2 40.9%

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

   7. Applied egg-rr 40.9%

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

      1. unpow1/2 40.9%

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

   9. Simplified 40.9%

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

   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 69.2%

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

      1. associate-*l/ 74.1%

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

   4. Applied egg-rr 74.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in l around inf 29.3%

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

      1. associate-/l* 29.2%

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

      2. associate-*r/ 29.2%

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

      3. metadata-eval 29.2%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in l around 0 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-*r* 24.1%

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

      3. associate-*l/ 24.0%

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

      4. fma-neg 24.0%

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

      5. associate-*r/ 24.0%

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

      6. metadata-eval 24.0%

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

      7. distribute-neg-frac 24.0%

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

      8. metadata-eval 24.0%

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

   10. Simplified 24.0%

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

   11. Taylor expanded in U around 0 31.8%

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

       1. *-commutative 31.8%

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

       2. associate-*r/ 31.8%

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

       3. metadata-eval 31.8%

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

   13. Simplified 31.8%

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

4. Final simplification 62.2%

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

Alternative 6: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(l_m \cdot l_m\right)}{Om}\right) + n \cdot \left({\left(\frac{l_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;l_m \leq 1.12 \cdot 10^{+226}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(l_m \cdot \frac{l_m}{Om}, -2, t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{Om} \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.1e+105)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+
       (- t (/ (* 2.0 (* l_m l_m)) Om))
       (* n (* (pow (/ l_m Om) 2.0) (- U* U)))))))
   (if (<= l_m 1.12e+226)
     (pow (* (* 2.0 (* n U)) (fma (* l_m (/ l_m Om)) -2.0 t)) 0.5)
     (* (* l_m (sqrt 2.0)) (sqrt (* (/ -2.0 Om) (* n U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.1e+105) {
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	} else if (l_m <= 1.12e+226) {
		tmp = pow(((2.0 * (n * U)) * fma((l_m * (l_m / Om)), -2.0, t)), 0.5);
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((-2.0 / Om) * (n * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.1e+105)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om)) + Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	elseif (l_m <= 1.12e+226)
		tmp = Float64(Float64(2.0 * Float64(n * U)) * fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)) ^ 0.5;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(-2.0 / Om) * Float64(n * U))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.1e+105], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.12e+226], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-2.0 / Om), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.10000000000000003e105

   1. Initial program 55.3%

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

   3. Simplified 52.3%

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

   if 1.10000000000000003e105 < l < 1.12e226

   1. Initial program 21.0%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in n around 0 21.3%

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

      1. pow1/2 31.8%

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

      2. associate-*r* 31.8%

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

      3. +-commutative 31.8%

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

      4. unpow2 31.8%

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

      5. associate-*l/ 51.6%

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

      6. *-commutative 51.6%

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

      7. fma-def 51.6%

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

      8. associate-*l/ 31.8%

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

      9. unpow2 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. unpow2 21.3%

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

      2. associate-*l/ 41.3%

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

   9. Applied egg-rr 51.6%

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

   if 1.12e226 < l

   1. Initial program 9.4%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in l around inf 27.9%

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

      1. associate-/l* 27.9%

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

      2. associate-*r/ 27.9%

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

      3. metadata-eval 27.9%

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

   7. Simplified 27.9%

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

   8. Taylor expanded in l around 0 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*r* 75.2%

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

      3. associate-*l/ 75.1%

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

      4. fma-neg 75.1%

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

      5. associate-*r/ 75.1%

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

      6. metadata-eval 75.1%

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

      7. distribute-neg-frac 75.1%

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

      8. metadata-eval 75.1%

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

   10. Simplified 75.1%

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

   11. Taylor expanded in n around 0 43.2%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+226}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{Om} \cdot \left(n \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(l_m \cdot \frac{l_m}{Om}, -2, t\right)\right)}^{0.5}\\ \mathbf{if}\;l_m \leq 8 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;l_m \leq 2.7 \cdot 10^{+152}:\\ \;\;\;\;{\left(\left(n \cdot {l_m}^{2}\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;l_m \leq 1.02 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{Om} \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (* (* 2.0 (* n U)) (fma (* l_m (/ l_m Om)) -2.0 t)) 0.5)))
   (if (<= l_m 8e+121)
     t_1
     (if (<= l_m 2.7e+152)
       (pow (* (* n (pow l_m 2.0)) (* -4.0 (/ U Om))) 0.5)
       (if (<= l_m 1.02e+225)
         t_1
         (* (* l_m (sqrt 2.0)) (sqrt (* (/ -2.0 Om) (* n U)))))))))

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(((2.0 * (n * U)) * fma((l_m * (l_m / Om)), -2.0, t)), 0.5);
	double tmp;
	if (l_m <= 8e+121) {
		tmp = t_1;
	} else if (l_m <= 2.7e+152) {
		tmp = pow(((n * pow(l_m, 2.0)) * (-4.0 * (U / Om))), 0.5);
	} else if (l_m <= 1.02e+225) {
		tmp = t_1;
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((-2.0 / Om) * (n * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * Float64(n * U)) * fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)) ^ 0.5
	tmp = 0.0
	if (l_m <= 8e+121)
		tmp = t_1;
	elseif (l_m <= 2.7e+152)
		tmp = Float64(Float64(n * (l_m ^ 2.0)) * Float64(-4.0 * Float64(U / Om))) ^ 0.5;
	elseif (l_m <= 1.02e+225)
		tmp = t_1;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(-2.0 / Om) * Float64(n * U))));
	end
	return 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[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[l$95$m, 8e+121], t$95$1, If[LessEqual[l$95$m, 2.7e+152], N[Power[N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 1.02e+225], t$95$1, N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-2.0 / Om), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.0000000000000003e121 or 2.70000000000000015e152 < l < 1.0200000000000001e225

   1. Initial program 51.6%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in n around 0 43.8%

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

      1. pow1/2 50.7%

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

      2. associate-*r* 50.7%

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

      3. +-commutative 50.7%

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

      4. unpow2 50.7%

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

      5. associate-*l/ 54.4%

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

      6. *-commutative 54.4%

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

      7. fma-def 54.4%

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

      8. associate-*l/ 50.7%

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

      9. unpow2 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. unpow2 43.8%

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

      2. associate-*l/ 47.5%

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

   9. Applied egg-rr 54.4%

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

   if 8.0000000000000003e121 < l < 2.70000000000000015e152

   1. Initial program 43.5%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in n around 0 43.4%

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

      1. pow1/2 61.6%

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

      2. associate-*r* 61.6%

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

      3. +-commutative 61.6%

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

      4. unpow2 61.6%

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

      5. associate-*l/ 61.8%

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

      6. *-commutative 61.8%

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

      7. fma-def 61.8%

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

      8. associate-*l/ 61.6%

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

      9. unpow2 61.6%

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

   7. Applied egg-rr 61.6%

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

   8. Taylor expanded in l around inf 73.8%

      \[\leadsto {\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}^{0.5} \]
   9. Step-by-step derivation

      1. *-commutative 73.8%

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

      2. *-commutative 73.8%

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

      3. associate-*l/ 82.5%

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

      4. *-commutative 82.5%

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

      5. associate-*l* 82.5%

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

   10. Simplified 82.5%

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

   if 1.0200000000000001e225 < l

   1. Initial program 9.4%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in l around inf 27.9%

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

      1. associate-/l* 27.9%

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

      2. associate-*r/ 27.9%

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

      3. metadata-eval 27.9%

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

   7. Simplified 27.9%

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

   8. Taylor expanded in l around 0 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*r* 75.2%

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

      3. associate-*l/ 75.1%

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

      4. fma-neg 75.1%

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

      5. associate-*r/ 75.1%

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

      6. metadata-eval 75.1%

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

      7. distribute-neg-frac 75.1%

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

      8. metadata-eval 75.1%

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

   10. Simplified 75.1%

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

   11. Taylor expanded in n around 0 43.2%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{+121}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+152}:\\ \;\;\;\;{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+225}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{Om} \cdot \left(n \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(l_m \cdot \frac{l_m}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{if}\;l_m \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;l_m \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(n \cdot {l_m}^{2}\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)}\\ \mathbf{elif}\;l_m \leq 8.5 \cdot 10^{+208}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{Om} \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* (* n U) (+ t (* (* l_m (/ l_m Om)) -2.0)))))))
   (if (<= l_m 9.5e+121)
     t_1
     (if (<= l_m 6e+153)
       (sqrt (* (* n (pow l_m 2.0)) (* -4.0 (/ U Om))))
       (if (<= l_m 8.5e+208)
         t_1
         (* (* l_m (sqrt 2.0)) (sqrt (* (/ -2.0 Om) (* n U)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	double tmp;
	if (l_m <= 9.5e+121) {
		tmp = t_1;
	} else if (l_m <= 6e+153) {
		tmp = sqrt(((n * pow(l_m, 2.0)) * (-4.0 * (U / Om))));
	} else if (l_m <= 8.5e+208) {
		tmp = t_1;
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((-2.0 / Om) * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * ((n * u) * (t + ((l_m * (l_m / om)) * (-2.0d0))))))
    if (l_m <= 9.5d+121) then
        tmp = t_1
    else if (l_m <= 6d+153) then
        tmp = sqrt(((n * (l_m ** 2.0d0)) * ((-4.0d0) * (u / om))))
    else if (l_m <= 8.5d+208) then
        tmp = t_1
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt((((-2.0d0) / om) * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	double tmp;
	if (l_m <= 9.5e+121) {
		tmp = t_1;
	} else if (l_m <= 6e+153) {
		tmp = Math.sqrt(((n * Math.pow(l_m, 2.0)) * (-4.0 * (U / Om))));
	} else if (l_m <= 8.5e+208) {
		tmp = t_1;
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((-2.0 / Om) * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))))
	tmp = 0
	if l_m <= 9.5e+121:
		tmp = t_1
	elif l_m <= 6e+153:
		tmp = math.sqrt(((n * math.pow(l_m, 2.0)) * (-4.0 * (U / Om))))
	elif l_m <= 8.5e+208:
		tmp = t_1
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((-2.0 / Om) * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * -2.0)))))
	tmp = 0.0
	if (l_m <= 9.5e+121)
		tmp = t_1;
	elseif (l_m <= 6e+153)
		tmp = sqrt(Float64(Float64(n * (l_m ^ 2.0)) * Float64(-4.0 * Float64(U / Om))));
	elseif (l_m <= 8.5e+208)
		tmp = t_1;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(-2.0 / Om) * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	tmp = 0.0;
	if (l_m <= 9.5e+121)
		tmp = t_1;
	elseif (l_m <= 6e+153)
		tmp = sqrt(((n * (l_m ^ 2.0)) * (-4.0 * (U / Om))));
	elseif (l_m <= 8.5e+208)
		tmp = t_1;
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((-2.0 / Om) * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l$95$m, 9.5e+121], t$95$1, If[LessEqual[l$95$m, 6e+153], N[Sqrt[N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 8.5e+208], t$95$1, N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-2.0 / Om), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.49999999999999949e121 or 6.00000000000000037e153 < l < 8.4999999999999992e208

   1. Initial program 53.2%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in n around 0 45.0%

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

      1. unpow2 45.0%

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

      2. associate-*l/ 48.1%

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

   7. Applied egg-rr 48.1%

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

   if 9.49999999999999949e121 < l < 6.00000000000000037e153

   1. Initial program 43.5%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in n around 0 43.4%

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

   6. Taylor expanded in t around 0 55.7%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

   8. Simplified 64.5%

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

      1. expm1-log1p-u 63.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n \cdot {\ell}^{2}}}\right)}\right)\right)} \]

      2. expm1-udef 32.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n \cdot {\ell}^{2}}}\right)}\right)} - 1} \]

      3. associate-*r* 32.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot -2\right) \cdot \frac{U}{\frac{Om}{n \cdot {\ell}^{2}}}}}\right)} - 1 \]

      4. metadata-eval 32.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{-4} \cdot \frac{U}{\frac{Om}{n \cdot {\ell}^{2}}}}\right)} - 1 \]

      5. associate-/r/ 32.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{-4 \cdot \color{blue}{\left(\frac{U}{Om} \cdot \left(n \cdot {\ell}^{2}\right)\right)}}\right)} - 1 \]

   10. Applied egg-rr 32.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{-4 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot {\ell}^{2}\right)\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 62.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-4 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot {\ell}^{2}\right)\right)}\right)\right)} \]

       2. expm1-log1p 64.4%

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

       3. *-commutative 64.4%

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

       4. *-commutative 64.4%

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

       5. associate-*l* 64.4%

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

   12. Simplified 64.4%

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

   if 8.4999999999999992e208 < l

   1. Initial program 6.3%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in l around inf 23.6%

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

      1. associate-/l* 23.6%

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

      2. associate-*r/ 23.6%

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

      3. metadata-eval 23.6%

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

   7. Simplified 23.6%

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

   8. Taylor expanded in l around 0 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-*r* 63.6%

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

      3. associate-*l/ 63.5%

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

      4. fma-neg 63.5%

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

      5. associate-*r/ 63.5%

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

      6. metadata-eval 63.5%

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

      7. distribute-neg-frac 63.5%

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

      8. metadata-eval 63.5%

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

   10. Simplified 63.5%

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

   11. Taylor expanded in n around 0 38.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(n \cdot {\ell}^{2}\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{Om} \cdot \left(n \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -4.8 \cdot 10^{-84} \lor \neg \left(Om \leq 3.7 \cdot 10^{-76}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(l_m \cdot \frac{l_m}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(n \cdot {l_m}^{2}\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -4.8e-84) (not (<= Om 3.7e-76)))
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l_m (/ l_m Om)) -2.0)))))
   (pow (* -4.0 (/ (* U (* n (pow l_m 2.0))) Om)) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -4.8e-84) || !(Om <= 3.7e-76)) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = pow((-4.0 * ((U * (n * pow(l_m, 2.0))) / Om)), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-4.8d-84)) .or. (.not. (om <= 3.7d-76))) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m * (l_m / om)) * (-2.0d0))))))
    else
        tmp = ((-4.0d0) * ((u * (n * (l_m ** 2.0d0))) / om)) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -4.8e-84) || !(Om <= 3.7e-76)) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = Math.pow((-4.0 * ((U * (n * Math.pow(l_m, 2.0))) / Om)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -4.8e-84) or not (Om <= 3.7e-76):
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))))
	else:
		tmp = math.pow((-4.0 * ((U * (n * math.pow(l_m, 2.0))) / Om)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -4.8e-84) || !(Om <= 3.7e-76))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * -2.0)))));
	else
		tmp = Float64(-4.0 * Float64(Float64(U * Float64(n * (l_m ^ 2.0))) / Om)) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -4.8e-84) || ~((Om <= 3.7e-76)))
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	else
		tmp = (-4.0 * ((U * (n * (l_m ^ 2.0))) / Om)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -4.8e-84], N[Not[LessEqual[Om, 3.7e-76]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.80000000000000035e-84 or 3.70000000000000011e-76 < Om

   1. Initial program 56.4%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in n around 0 50.7%

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

      1. unpow2 50.7%

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

      2. associate-*l/ 56.0%

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

   7. Applied egg-rr 56.0%

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

   if -4.80000000000000035e-84 < Om < 3.70000000000000011e-76

   1. Initial program 32.0%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in n around 0 21.7%

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

      1. pow1/2 41.8%

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

      2. associate-*r* 41.8%

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

      3. +-commutative 41.8%

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

      4. unpow2 41.8%

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

      5. associate-*l/ 41.8%

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

      6. *-commutative 41.8%

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

      7. fma-def 41.8%

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

      8. associate-*l/ 41.8%

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

      9. unpow2 41.8%

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

   7. Applied egg-rr 41.8%

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

   8. Taylor expanded in l around inf 36.6%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.8 \cdot 10^{-84} \lor \neg \left(Om \leq 3.7 \cdot 10^{-76}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{-84} \lor \neg \left(Om \leq 9 \cdot 10^{-74}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(l_m \cdot \frac{l_m}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot {l_m}^{2}\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -2.5e-84) (not (<= Om 9e-74)))
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l_m (/ l_m Om)) -2.0)))))
   (pow (* (* n (pow l_m 2.0)) (* -4.0 (/ U Om))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.5e-84) || !(Om <= 9e-74)) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = pow(((n * pow(l_m, 2.0)) * (-4.0 * (U / Om))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-2.5d-84)) .or. (.not. (om <= 9d-74))) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m * (l_m / om)) * (-2.0d0))))))
    else
        tmp = ((n * (l_m ** 2.0d0)) * ((-4.0d0) * (u / om))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.5e-84) || !(Om <= 9e-74)) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = Math.pow(((n * Math.pow(l_m, 2.0)) * (-4.0 * (U / Om))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -2.5e-84) or not (Om <= 9e-74):
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))))
	else:
		tmp = math.pow(((n * math.pow(l_m, 2.0)) * (-4.0 * (U / Om))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -2.5e-84) || !(Om <= 9e-74))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * -2.0)))));
	else
		tmp = Float64(Float64(n * (l_m ^ 2.0)) * Float64(-4.0 * Float64(U / Om))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -2.5e-84) || ~((Om <= 9e-74)))
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	else
		tmp = ((n * (l_m ^ 2.0)) * (-4.0 * (U / Om))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -2.5e-84], N[Not[LessEqual[Om, 9e-74]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.5000000000000001e-84 or 8.9999999999999998e-74 < Om

   1. Initial program 56.4%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in n around 0 50.7%

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

      1. unpow2 50.7%

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

      2. associate-*l/ 56.0%

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

   7. Applied egg-rr 56.0%

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

   if -2.5000000000000001e-84 < Om < 8.9999999999999998e-74

   1. Initial program 32.0%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in n around 0 21.7%

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

      1. pow1/2 41.8%

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

      2. associate-*r* 41.8%

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

      3. +-commutative 41.8%

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

      4. unpow2 41.8%

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

      5. associate-*l/ 41.8%

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

      6. *-commutative 41.8%

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

      7. fma-def 41.8%

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

      8. associate-*l/ 41.8%

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

      9. unpow2 41.8%

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

   7. Applied egg-rr 41.8%

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

   8. Taylor expanded in l around inf 36.6%

      \[\leadsto {\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}^{0.5} \]
   9. Step-by-step derivation

      1. *-commutative 36.6%

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

      2. *-commutative 36.6%

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

      3. associate-*l/ 39.3%

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

      4. *-commutative 39.3%

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

      5. associate-*l* 39.3%

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

   10. Simplified 39.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{-84} \lor \neg \left(Om \leq 9 \cdot 10^{-74}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(l_m \cdot \frac{l_m}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 5.1e+60)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l_m (/ l_m Om)) -2.0)))))
   (* (sqrt (* n (* 2.0 U))) (sqrt 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 (t <= 5.1e+60) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(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 (t <= 5.1d+60) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m * (l_m / om)) * (-2.0d0))))))
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(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 (t <= 5.1e+60) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 5.1e+60:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))))
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 5.1e+60)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * -2.0)))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(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 (t <= 5.1e+60)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(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[t, 5.1e+60], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.09999999999999996e60

   1. Initial program 50.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in n around 0 41.6%

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

      1. unpow2 41.6%

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

      2. associate-*l/ 46.7%

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

   7. Applied egg-rr 46.7%

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

   if 5.09999999999999996e60 < t

   1. Initial program 44.9%

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

      1. associate-*l/ 45.1%

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

   4. Applied egg-rr 45.1%

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

   5. Taylor expanded in t around inf 44.0%

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

      1. sqrt-prod 56.0%

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

      2. *-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. *-commutative 56.0%

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

      3. associate-*l* 56.0%

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

   9. Simplified 56.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 1.05 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(l_m \cdot \frac{l_m}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l_m \cdot \sqrt{n \cdot \frac{U \cdot -4}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.05e+209)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l_m (/ l_m Om)) -2.0)))))
   (* l_m (sqrt (* n (/ (* U -4.0) 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 <= 1.05e+209) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = l_m * sqrt((n * ((U * -4.0) / 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 <= 1.05d+209) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m * (l_m / om)) * (-2.0d0))))))
    else
        tmp = l_m * sqrt((n * ((u * (-4.0d0)) / 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 <= 1.05e+209) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = l_m * Math.sqrt((n * ((U * -4.0) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.05e+209:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))))
	else:
		tmp = l_m * math.sqrt((n * ((U * -4.0) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.05e+209)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * -2.0)))));
	else
		tmp = Float64(l_m * sqrt(Float64(n * Float64(Float64(U * -4.0) / 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 <= 1.05e+209)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	else
		tmp = l_m * sqrt((n * ((U * -4.0) / 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, 1.05e+209], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05e209

   1. Initial program 52.7%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in n around 0 45.0%

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

      1. unpow2 45.0%

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

      2. associate-*l/ 47.9%

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

   7. Applied egg-rr 47.9%

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

   if 1.05e209 < l

   1. Initial program 6.3%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in l around inf 23.6%

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

      1. associate-/l* 23.6%

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

      2. associate-*r/ 23.6%

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

      3. metadata-eval 23.6%

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

   7. Simplified 23.6%

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

   8. Taylor expanded in l around 0 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-*r* 63.6%

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

      3. associate-*l/ 63.5%

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

      4. fma-neg 63.5%

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

      5. associate-*r/ 63.5%

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

      6. metadata-eval 63.5%

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

      7. distribute-neg-frac 63.5%

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

      8. metadata-eval 63.5%

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

   10. Simplified 63.5%

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

   11. Taylor expanded in n around 0 38.1%

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

       1. associate-/l* 41.8%

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

   13. Simplified 41.8%

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

       1. expm1-log1p-u 39.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)\right)} \]

       2. expm1-udef 39.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)} - 1} \]

       3. associate-*l* 39.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)}\right)} - 1 \]

       4. pow1/2 39.0%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)\right)} - 1 \]

       5. pow1/2 49.5%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \left({2}^{0.5} \cdot \color{blue}{{\left(-2 \cdot \frac{U}{\frac{Om}{n}}\right)}^{0.5}}\right)\right)} - 1 \]

       6. pow-prod-down 49.5%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{\left(2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n}}\right)\right)}^{0.5}}\right)} - 1 \]

       7. associate-/r/ 44.9%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot {\left(2 \cdot \left(-2 \cdot \color{blue}{\left(\frac{U}{Om} \cdot n\right)}\right)\right)}^{0.5}\right)} - 1 \]

   15. Applied egg-rr 44.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot {\left(2 \cdot \left(-2 \cdot \left(\frac{U}{Om} \cdot n\right)\right)\right)}^{0.5}\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 44.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {\left(2 \cdot \left(-2 \cdot \left(\frac{U}{Om} \cdot n\right)\right)\right)}^{0.5}\right)\right)} \]

       2. expm1-log1p 47.5%

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

       3. unpow1/2 37.0%

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

       4. associate-*r* 37.0%

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

       5. metadata-eval 37.0%

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

       6. associate-*r* 37.0%

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

       7. associate-*r/ 37.0%

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

   17. Simplified 37.0%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \frac{U \cdot -4}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;l_m \cdot \sqrt{n \cdot \frac{U \cdot -4}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.4e+82)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (* l_m (sqrt (* n (/ (* U -4.0) 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 <= 5.4e+82) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = l_m * sqrt((n * ((U * -4.0) / 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 <= 5.4d+82) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = l_m * sqrt((n * ((u * (-4.0d0)) / 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 <= 5.4e+82) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = l_m * Math.sqrt((n * ((U * -4.0) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.4e+82:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = l_m * math.sqrt((n * ((U * -4.0) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.4e+82)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = Float64(l_m * sqrt(Float64(n * Float64(Float64(U * -4.0) / 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 <= 5.4e+82)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = l_m * sqrt((n * ((U * -4.0) / 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, 5.4e+82], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.3999999999999999e82

   1. Initial program 54.5%

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

      1. associate-*l/ 55.9%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in t around inf 38.9%

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

      1. pow1/2 41.3%

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

      2. *-commutative 41.3%

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

   7. Applied egg-rr 41.3%

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

   if 5.3999999999999999e82 < l

   1. Initial program 26.1%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in l around inf 35.1%

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

      1. associate-/l* 35.0%

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

      2. associate-*r/ 35.0%

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

      3. metadata-eval 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in l around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*r* 55.3%

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

      3. associate-*l/ 55.3%

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

      4. fma-neg 55.3%

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

      5. associate-*r/ 55.3%

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

      6. metadata-eval 55.3%

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

      7. distribute-neg-frac 55.3%

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

      8. metadata-eval 55.3%

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

   10. Simplified 55.3%

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

   11. Taylor expanded in n around 0 31.4%

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

       1. associate-/l* 37.0%

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

   13. Simplified 37.0%

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

       1. expm1-log1p-u 35.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)\right)} \]

       2. expm1-udef 31.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)} - 1} \]

       3. associate-*l* 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)}\right)} - 1 \]

       4. pow1/2 31.7%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)\right)} - 1 \]

       5. pow1/2 38.1%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \left({2}^{0.5} \cdot \color{blue}{{\left(-2 \cdot \frac{U}{\frac{Om}{n}}\right)}^{0.5}}\right)\right)} - 1 \]

       6. pow-prod-down 38.1%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{\left(2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n}}\right)\right)}^{0.5}}\right)} - 1 \]

       7. associate-/r/ 35.1%

          \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot {\left(2 \cdot \left(-2 \cdot \color{blue}{\left(\frac{U}{Om} \cdot n\right)}\right)\right)}^{0.5}\right)} - 1 \]

   15. Applied egg-rr 35.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot {\left(2 \cdot \left(-2 \cdot \left(\frac{U}{Om} \cdot n\right)\right)\right)}^{0.5}\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 39.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {\left(2 \cdot \left(-2 \cdot \left(\frac{U}{Om} \cdot n\right)\right)\right)}^{0.5}\right)\right)} \]

       2. expm1-log1p 41.4%

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

       3. unpow1/2 37.2%

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

       4. associate-*r* 37.2%

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

       5. metadata-eval 37.2%

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

       6. associate-*r* 37.2%

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

       7. associate-*r/ 37.2%

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

   17. Simplified 37.2%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \frac{U \cdot -4}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 3.4 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.4e-116)
   (sqrt (* 2.0 (* t (* n U))))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.4e-116) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 3.4d-116) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.4e-116) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 3.4e-116:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.4e-116)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 3.4e-116)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.4e-116], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.39999999999999992e-116

   1. Initial program 53.9%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in l around 0 40.3%

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

   if 3.39999999999999992e-116 < l

   1. Initial program 41.9%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in n around 0 36.2%

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

      1. pow1/2 45.5%

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

      2. associate-*r* 45.5%

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

      3. +-commutative 45.5%

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

      4. unpow2 45.5%

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

      5. associate-*l/ 52.3%

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

      6. *-commutative 52.3%

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

      7. fma-def 52.3%

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

      8. associate-*l/ 45.5%

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

      9. unpow2 45.5%

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

   7. Applied egg-rr 45.5%

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

   8. Taylor expanded in l around 0 24.9%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.55e+77)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.55e+77) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.55d+77) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.55e+77) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.55e+77:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.55e+77)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.55e+77)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.55e+77], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.54999999999999999e77

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in l around 0 35.9%

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

      1. pow1/2 38.4%

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

      2. associate-*r* 41.2%

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

      3. *-commutative 41.2%

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

   7. Applied egg-rr 41.2%

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

   if 1.54999999999999999e77 < l

   1. Initial program 26.1%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in n around 0 22.4%

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

      1. pow1/2 33.2%

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

      2. associate-*r* 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l/ 47.5%

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

      6. *-commutative 47.5%

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

      7. fma-def 47.5%

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

      8. associate-*l/ 33.2%

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

      9. unpow2 33.2%

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

   7. Applied egg-rr 33.2%

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

   8. Taylor expanded in l around 0 13.2%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.2e+77)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.2e+77) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 5.2d+77) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.2e+77) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.2e+77:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.2e+77)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 5.2e+77)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5.2e+77], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.2000000000000004e77

   1. Initial program 54.5%

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

      1. associate-*l/ 55.9%

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

   4. Applied egg-rr 55.9%

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

   5. Taylor expanded in t around inf 38.9%

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

      1. pow1/2 41.3%

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

      2. *-commutative 41.3%

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

   7. Applied egg-rr 41.3%

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

   if 5.2000000000000004e77 < l

   1. Initial program 26.1%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in n around 0 22.4%

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

      1. pow1/2 33.2%

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

      2. associate-*r* 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l/ 47.5%

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

      6. *-commutative 47.5%

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

      7. fma-def 47.5%

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

      8. associate-*l/ 33.2%

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

      9. unpow2 33.2%

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

   7. Applied egg-rr 33.2%

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

   8. Taylor expanded in l around 0 13.2%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

3. Simplified 51.9%

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

5. Taylor expanded in l around 0 31.4%

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

6. Final simplification 31.4%

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

Alternative 18: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

3. Simplified 51.9%

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

5. Taylor expanded in l around 0 33.3%

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

6. Final simplification 33.3%

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

Alternative 19: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

   1. associate-*l/ 53.1%

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

4. Applied egg-rr 53.1%

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

5. Taylor expanded in t around inf 33.4%

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

6. Final simplification 33.4%

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

Reproduce

herbie shell --seed 2024018 
(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*))))))