Toniolo and Linder, Equation (13)

Percentage Accurate: 49.7% → 67.4%

Time: 26.2s

Alternatives: 18

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 11.0%

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

   3. Simplified 32.9%

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

      1. *-commutative 32.9%

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

      2. sqrt-prod 45.9%

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

   6. Applied egg-rr 43.0%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 71.4%

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

   3. Simplified 77.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 14.2%

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

   5. Taylor expanded in U around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. associate-/l* 0.6%

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

      3. unpow2 0.6%

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

      4. unpow2 0.6%

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

      5. times-frac 14.3%

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

      6. unpow2 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in Om around -inf 17.1%

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

      1. mul-1-neg 17.1%

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

      2. fma-define 17.1%

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

      3. associate-/l* 17.0%

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

   10. Simplified 17.0%

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

   11. Taylor expanded in l around 0 32.3%

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

       1. associate-/l* 32.3%

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

       2. sub-neg 32.3%

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

       3. associate-/l* 34.9%

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

       4. metadata-eval 34.9%

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

   13. Simplified 34.9%

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

   14. Taylor expanded in t around 0 14.3%

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

       1. associate-/l* 20.2%

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

       2. associate-*r/ 20.2%

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

       3. fmm-def 20.2%

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

       4. metadata-eval 20.2%

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

   16. Simplified 20.2%

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

4. Final simplification 64.2%

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

Alternative 2: 67.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 11.0%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in Om around inf 30.1%

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

      1. pow1/2 30.1%

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

      2. *-commutative 30.1%

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

      3. unpow-prod-down 40.6%

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

      4. pow1/2 40.6%

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

      5. cancel-sign-sub-inv 40.6%

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

      6. metadata-eval 40.6%

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

      7. pow1/2 40.6%

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

   7. Applied egg-rr 40.6%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 71.4%

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

   3. Simplified 77.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 14.2%

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

   5. Taylor expanded in U around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. associate-/l* 0.6%

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

      3. unpow2 0.6%

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

      4. unpow2 0.6%

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

      5. times-frac 14.3%

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

      6. unpow2 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in Om around -inf 17.1%

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

      1. mul-1-neg 17.1%

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

      2. fma-define 17.1%

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

      3. associate-/l* 17.0%

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

   10. Simplified 17.0%

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

   11. Taylor expanded in l around 0 32.3%

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

       1. associate-/l* 32.3%

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

       2. sub-neg 32.3%

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

       3. associate-/l* 34.9%

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

       4. metadata-eval 34.9%

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

   13. Simplified 34.9%

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

   14. Taylor expanded in t around 0 14.3%

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

       1. associate-/l* 20.2%

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

       2. associate-*r/ 20.2%

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

       3. fmm-def 20.2%

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

       4. metadata-eval 20.2%

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

   16. Simplified 20.2%

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

4. Final simplification 63.9%

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

Alternative 3: 67.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{{l\_m}^{2}}{Om} \cdot -2\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* (/ (pow l_m 2.0) Om) -2.0)))))
     (if (<= t_1 5e+152)
       t_1
       (*
        (* l_m (sqrt 2.0))
        (sqrt (/ (* U (* n (- (/ (* n U*) Om) 2.0))) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + ((pow(l_m, 2.0) / Om) * -2.0))));
	} else if (t_1 <= 5e+152) {
		tmp = t_1;
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + (((l_m ** 2.0d0) / om) * (-2.0d0)))))
    else if (t_1 <= 5d+152) then
        tmp = t_1
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((u * (n * (((n * u_42) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + ((Math.pow(l_m, 2.0) / Om) * -2.0))));
	} else if (t_1 <= 5e+152) {
		tmp = t_1;
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	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 - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + ((math.pow(l_m, 2.0) / Om) * -2.0))))
	elif t_1 <= 5e+152:
		tmp = t_1
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(Float64((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_1 <= 5e+152)
		tmp = t_1;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (((l_m ^ 2.0) / Om) * -2.0))));
	elseif (t_1 <= 5e+152)
		tmp = t_1;
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], t$95$1, N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 11.0%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in Om around inf 30.1%

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

      1. pow1/2 30.1%

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

      2. *-commutative 30.1%

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

      3. unpow-prod-down 40.6%

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

      4. pow1/2 40.6%

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

      5. cancel-sign-sub-inv 40.6%

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

      6. metadata-eval 40.6%

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

      7. pow1/2 40.6%

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

   7. Applied egg-rr 40.6%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.3%

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

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 22.3%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in U around 0 23.2%

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

      1. mul-1-neg 23.2%

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

      2. associate-/l* 19.9%

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

      3. unpow2 19.9%

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

      4. unpow2 19.9%

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

      5. times-frac 34.6%

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

      6. unpow2 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in Om around -inf 26.6%

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

      1. mul-1-neg 26.6%

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

      2. fma-define 26.6%

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

      3. associate-/l* 26.6%

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

   10. Simplified 26.6%

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

   11. Taylor expanded in l around 0 31.4%

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

       1. associate-/l* 31.3%

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

       2. sub-neg 31.3%

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

       3. associate-/l* 32.4%

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

       4. metadata-eval 32.4%

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

   13. Simplified 32.4%

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

   14. Taylor expanded in t around 0 24.6%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\\ t_2 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - t\_1\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* l_m (/ l_m Om))))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2))))
   (if (<= t_3 2e-217)
     (sqrt (* (* 2.0 n) (* U (- t t_1))))
     (if (<= t_3 2e+305)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 t_1))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (/ (* U (* n (- (/ (* n U*) Om) 2.0))) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (l_m * (l_m / Om));
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double tmp;
	if (t_3 <= 2e-217) {
		tmp = sqrt(((2.0 * n) * (U * (t - t_1))));
	} else if (t_3 <= 2e+305) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 * (l_m * (l_m / om))
    t_2 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_3 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_2)
    if (t_3 <= 2d-217) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - t_1))))
    else if (t_3 <= 2d+305) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_2 - t_1))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((u * (n * (((n * u_42) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (l_m * (l_m / Om));
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double tmp;
	if (t_3 <= 2e-217) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - t_1))));
	} else if (t_3 <= 2e+305) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (l_m * (l_m / Om))
	t_2 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)
	tmp = 0
	if t_3 <= 2e-217:
		tmp = math.sqrt(((2.0 * n) * (U * (t - t_1))))
	elif t_3 <= 2e+305:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(l_m * Float64(l_m / Om)))
	t_2 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2))
	tmp = 0.0
	if (t_3 <= 2e-217)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - t_1))));
	elseif (t_3 <= 2e+305)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - t_1))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (l_m * (l_m / Om));
	t_2 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	tmp = 0.0;
	if (t_3 <= 2e-217)
		tmp = sqrt(((2.0 * n) * (U * (t - t_1))));
	elseif (t_3 <= 2e+305)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-217], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 2e+305], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000016e-217

   1. Initial program 27.9%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in Om around inf 44.3%

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

      1. unpow2 44.3%

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

      2. associate-*r/ 46.3%

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

      3. *-commutative 46.3%

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

   7. Applied egg-rr 46.3%

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

   if 2.00000000000000016e-217 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.9999999999999999e305

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   if 1.9999999999999999e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 23.8%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in U around 0 21.0%

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

      1. mul-1-neg 21.0%

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

      2. associate-/l* 21.2%

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

      3. unpow2 21.2%

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

      4. unpow2 21.2%

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

      5. times-frac 35.9%

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

      6. unpow2 35.9%

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

   7. Simplified 35.9%

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

   8. Taylor expanded in Om around -inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. fma-define 24.6%

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

      3. associate-/l* 24.6%

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

   10. Simplified 24.6%

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

   11. Taylor expanded in l around 0 31.5%

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

       1. associate-/l* 31.4%

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

       2. sub-neg 31.4%

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

       3. associate-/l* 31.7%

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

       4. metadata-eval 31.7%

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

   13. Simplified 31.7%

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

   14. Taylor expanded in t around 0 25.1%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l\_m \cdot \frac{l\_m}{Om}\\ \mathbf{if}\;l\_m \leq 3.1 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 2.35 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + -2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om))))
   (if (<= l_m 3.1e-180)
     (sqrt (fabs (* (* (* 2.0 n) U) t)))
     (if (<= l_m 2e+144)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (* (* l_m l_m) (/ (+ -2.0 (* U* (/ n Om))) Om))))))
       (if (<= l_m 2.35e+226)
         (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1)))))
         (* (sqrt (* 2.0 (* n U))) (sqrt (+ t (* -2.0 t_1)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double tmp;
	if (l_m <= 3.1e-180) {
		tmp = sqrt(fabs((((2.0 * n) * U) * t)));
	} else if (l_m <= 2e+144) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else if (l_m <= 2.35e+226) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt((t + (-2.0 * t_1)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l_m * (l_m / om)
    if (l_m <= 3.1d-180) then
        tmp = sqrt(abs((((2.0d0 * n) * u) * t)))
    else if (l_m <= 2d+144) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l_m * l_m) * (((-2.0d0) + (u_42 * (n / om))) / om))))))
    else if (l_m <= 2.35d+226) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt((t + ((-2.0d0) * t_1)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double tmp;
	if (l_m <= 3.1e-180) {
		tmp = Math.sqrt(Math.abs((((2.0 * n) * U) * t)));
	} else if (l_m <= 2e+144) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else if (l_m <= 2.35e+226) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt((t + (-2.0 * t_1)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	tmp = 0
	if l_m <= 3.1e-180:
		tmp = math.sqrt(math.fabs((((2.0 * n) * U) * t)))
	elif l_m <= 2e+144:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))))
	elif l_m <= 2.35e+226:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt((t + (-2.0 * t_1)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m * Float64(l_m / Om))
	tmp = 0.0
	if (l_m <= 3.1e-180)
		tmp = sqrt(abs(Float64(Float64(Float64(2.0 * n) * U) * t)));
	elseif (l_m <= 2e+144)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(-2.0 + Float64(U_42_ * Float64(n / Om))) / Om))))));
	elseif (l_m <= 2.35e+226)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(Float64(t + Float64(-2.0 * t_1))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = l_m * (l_m / Om);
	tmp = 0.0;
	if (l_m <= 3.1e-180)
		tmp = sqrt(abs((((2.0 * n) * U) * t)));
	elseif (l_m <= 2e+144)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	elseif (l_m <= 2.35e+226)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt((t + (-2.0 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 3.1e-180], N[Sqrt[N[Abs[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2e+144], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.35e+226], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.0999999999999999e-180

   1. Initial program 57.0%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in l around 0 42.7%

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

      1. associate-*r* 42.7%

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

   7. Simplified 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. pow1/2 42.7%

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

      3. pow1/2 43.4%

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

      4. pow-prod-down 30.5%

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

      5. pow2 30.5%

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

   9. Applied egg-rr 30.5%

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

       1. unpow1/2 30.5%

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

       2. unpow2 30.5%

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

       3. rem-sqrt-square 43.9%

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

       4. associate-*r* 48.5%

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

       5. associate-*r* 48.5%

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

       6. *-commutative 48.5%

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

       7. associate-*r* 48.5%

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

   11. Simplified 48.5%

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

   if 3.0999999999999999e-180 < l < 2.00000000000000005e144

   1. Initial program 48.8%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in U around 0 50.2%

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

      1. mul-1-neg 50.2%

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

      2. associate-/l* 50.1%

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

      3. unpow2 50.1%

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

      4. unpow2 50.1%

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

      5. times-frac 50.1%

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

      6. unpow2 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in Om around -inf 53.3%

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

      1. mul-1-neg 53.3%

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

      2. fma-define 53.3%

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

      3. associate-/l* 56.3%

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

   10. Simplified 56.3%

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

   11. Taylor expanded in l around 0 50.5%

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

       1. associate-/l* 50.4%

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

       2. sub-neg 50.4%

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

       3. associate-/l* 57.9%

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

       4. metadata-eval 57.9%

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

   13. Simplified 57.9%

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

       1. unpow2 57.9%

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

   15. Applied egg-rr 57.9%

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

   if 2.00000000000000005e144 < l < 2.34999999999999996e226

   1. Initial program 34.2%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in Om around inf 34.7%

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

      1. unpow2 34.7%

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

      2. associate-*r/ 66.9%

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

      3. *-commutative 66.9%

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

   7. Applied egg-rr 66.9%

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

   if 2.34999999999999996e226 < l

   1. Initial program 29.6%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in Om around inf 29.6%

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

      1. pow1/2 48.1%

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

      2. associate-*r* 48.1%

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

      3. unpow-prod-down 39.0%

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

      4. pow1/2 29.6%

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

      5. cancel-sign-sub-inv 29.6%

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

      6. metadata-eval 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. unpow1/2 29.6%

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

      2. associate-*l* 29.6%

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

   9. Simplified 29.6%

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

       1. unpow2 29.6%

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

       2. associate-*r/ 38.5%

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

       3. *-commutative 38.5%

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

   11. Applied egg-rr 81.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 2.9 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(\frac{{l\_m}^{2}}{Om} \cdot \left(U \cdot -2\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.4e-180)
   (sqrt (fabs (* (* (* 2.0 n) U) t)))
   (if (<= l_m 2e+144)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (* l_m l_m) (/ (+ -2.0 (* U* (/ n Om))) Om))))))
     (if (<= l_m 2.9e+214)
       (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l_m (/ l_m Om)))))))
       (pow (* (* 2.0 n) (* (/ (pow l_m 2.0) Om) (* U -2.0))) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.4e-180) {
		tmp = sqrt(fabs((((2.0 * n) * U) * t)));
	} else if (l_m <= 2e+144) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else if (l_m <= 2.9e+214) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = pow(((2.0 * n) * ((pow(l_m, 2.0) / Om) * (U * -2.0))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 4.4d-180) then
        tmp = sqrt(abs((((2.0d0 * n) * u) * t)))
    else if (l_m <= 2d+144) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l_m * l_m) * (((-2.0d0) + (u_42 * (n / om))) / om))))))
    else if (l_m <= 2.9d+214) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = ((2.0d0 * n) * (((l_m ** 2.0d0) / om) * (u * (-2.0d0)))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.4e-180) {
		tmp = Math.sqrt(Math.abs((((2.0 * n) * U) * t)));
	} else if (l_m <= 2e+144) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else if (l_m <= 2.9e+214) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.pow(((2.0 * n) * ((Math.pow(l_m, 2.0) / Om) * (U * -2.0))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 4.4e-180:
		tmp = math.sqrt(math.fabs((((2.0 * n) * U) * t)))
	elif l_m <= 2e+144:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))))
	elif l_m <= 2.9e+214:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.pow(((2.0 * n) * ((math.pow(l_m, 2.0) / Om) * (U * -2.0))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4.4e-180)
		tmp = sqrt(abs(Float64(Float64(Float64(2.0 * n) * U) * t)));
	elseif (l_m <= 2e+144)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(-2.0 + Float64(U_42_ * Float64(n / Om))) / Om))))));
	elseif (l_m <= 2.9e+214)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(Float64((l_m ^ 2.0) / Om) * Float64(U * -2.0))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 4.4e-180)
		tmp = sqrt(abs((((2.0 * n) * U) * t)));
	elseif (l_m <= 2e+144)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	elseif (l_m <= 2.9e+214)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = ((2.0 * n) * (((l_m ^ 2.0) / Om) * (U * -2.0))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 4.4e-180], N[Sqrt[N[Abs[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2e+144], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.9e+214], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 4.40000000000000026e-180

   1. Initial program 57.0%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in l around 0 42.7%

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

      1. associate-*r* 42.7%

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

   7. Simplified 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. pow1/2 42.7%

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

      3. pow1/2 43.4%

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

      4. pow-prod-down 30.5%

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

      5. pow2 30.5%

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

   9. Applied egg-rr 30.5%

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

       1. unpow1/2 30.5%

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

       2. unpow2 30.5%

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

       3. rem-sqrt-square 43.9%

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

       4. associate-*r* 48.5%

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

       5. associate-*r* 48.5%

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

       6. *-commutative 48.5%

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

       7. associate-*r* 48.5%

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

   11. Simplified 48.5%

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

   if 4.40000000000000026e-180 < l < 2.00000000000000005e144

   1. Initial program 48.8%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in U around 0 50.2%

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

      1. mul-1-neg 50.2%

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

      2. associate-/l* 50.1%

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

      3. unpow2 50.1%

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

      4. unpow2 50.1%

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

      5. times-frac 50.1%

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

      6. unpow2 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in Om around -inf 53.3%

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

      1. mul-1-neg 53.3%

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

      2. fma-define 53.3%

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

      3. associate-/l* 56.3%

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

   10. Simplified 56.3%

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

   11. Taylor expanded in l around 0 50.5%

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

       1. associate-/l* 50.4%

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

       2. sub-neg 50.4%

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

       3. associate-/l* 57.9%

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

       4. metadata-eval 57.9%

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

   13. Simplified 57.9%

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

       1. unpow2 57.9%

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

   15. Applied egg-rr 57.9%

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

   if 2.00000000000000005e144 < l < 2.8999999999999999e214

   1. Initial program 50.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in Om around inf 50.8%

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

      1. unpow2 50.8%

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

      2. associate-*r/ 83.1%

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

      3. *-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 2.8999999999999999e214 < l

   1. Initial program 23.5%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in Om around inf 23.8%

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

   6. Taylor expanded in t around 0 23.8%

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

      1. associate-*r/ 23.8%

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

      2. associate-*r* 23.8%

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

   8. Simplified 23.8%

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

      1. pow1/2 45.5%

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

      2. associate-/l* 45.5%

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

   10. Applied egg-rr 45.5%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \left(U \cdot -2\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;l\_m \leq 1.7 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 7e-179)
   (sqrt (fabs (* (* (* 2.0 n) U) t)))
   (if (<= l_m 1.7e+156)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (* l_m l_m) (/ (+ -2.0 (* U* (/ n Om))) Om))))))
     (*
      (* l_m (sqrt 2.0))
      (sqrt (/ (* U (* n (- (/ (* n U*) Om) 2.0))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 7e-179) {
		tmp = sqrt(fabs((((2.0 * n) * U) * t)));
	} else if (l_m <= 1.7e+156) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 7d-179) then
        tmp = sqrt(abs((((2.0d0 * n) * u) * t)))
    else if (l_m <= 1.7d+156) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l_m * l_m) * (((-2.0d0) + (u_42 * (n / om))) / om))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((u * (n * (((n * u_42) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 7e-179) {
		tmp = Math.sqrt(Math.abs((((2.0 * n) * U) * t)));
	} else if (l_m <= 1.7e+156) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 7e-179:
		tmp = math.sqrt(math.fabs((((2.0 * n) * U) * t)))
	elif l_m <= 1.7e+156:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 7e-179)
		tmp = sqrt(abs(Float64(Float64(Float64(2.0 * n) * U) * t)));
	elseif (l_m <= 1.7e+156)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(-2.0 + Float64(U_42_ * Float64(n / Om))) / Om))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 7e-179)
		tmp = sqrt(abs((((2.0 * n) * U) * t)));
	elseif (l_m <= 1.7e+156)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 7e-179], N[Sqrt[N[Abs[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.7e+156], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.00000000000000049e-179

   1. Initial program 57.0%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in l around 0 42.7%

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

      1. associate-*r* 42.7%

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

   7. Simplified 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. pow1/2 42.7%

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

      3. pow1/2 43.4%

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

      4. pow-prod-down 30.5%

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

      5. pow2 30.5%

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

   9. Applied egg-rr 30.5%

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

       1. unpow1/2 30.5%

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

       2. unpow2 30.5%

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

       3. rem-sqrt-square 43.9%

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

       4. associate-*r* 48.5%

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

       5. associate-*r* 48.5%

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

       6. *-commutative 48.5%

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

       7. associate-*r* 48.5%

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

   11. Simplified 48.5%

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

   if 7.00000000000000049e-179 < l < 1.7e156

   1. Initial program 50.3%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in U around 0 50.2%

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

      1. mul-1-neg 50.2%

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

      2. associate-/l* 51.5%

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

      3. unpow2 51.5%

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

      4. unpow2 51.5%

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

      5. times-frac 51.5%

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

      6. unpow2 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in Om around -inf 53.2%

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

      1. mul-1-neg 53.2%

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

      2. fma-define 53.2%

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

      3. associate-/l* 57.5%

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

   10. Simplified 57.5%

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

   11. Taylor expanded in l around 0 51.9%

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

       1. associate-/l* 51.8%

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

       2. sub-neg 51.8%

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

       3. associate-/l* 59.1%

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

       4. metadata-eval 59.1%

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

   13. Simplified 59.1%

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

       1. unpow2 59.1%

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

   15. Applied egg-rr 59.1%

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

   if 1.7e156 < l

   1. Initial program 24.1%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in U around 0 16.9%

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

      1. mul-1-neg 16.9%

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

      2. associate-/l* 16.9%

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

      3. unpow2 16.9%

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

      4. unpow2 16.9%

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

      5. times-frac 45.9%

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

      6. unpow2 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in Om around -inf 23.9%

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

      1. mul-1-neg 23.9%

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

      2. fma-define 23.9%

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

      3. associate-/l* 23.9%

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

   10. Simplified 23.9%

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

   11. Taylor expanded in l around 0 41.3%

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

       1. associate-/l* 41.3%

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

       2. sub-neg 41.3%

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

       3. associate-/l* 41.3%

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

       4. metadata-eval 41.3%

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

   13. Simplified 41.3%

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

   14. Taylor expanded in t around 0 72.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+144} \lor \neg \left(l\_m \leq 3.25 \cdot 10^{+214}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8e-180)
   (sqrt (fabs (* (* (* 2.0 n) U) t)))
   (if (or (<= l_m 2e+144) (not (<= l_m 3.25e+214)))
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (* l_m l_m) (/ (+ -2.0 (* U* (/ n Om))) Om))))))
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l_m (/ l_m 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 <= 8e-180) {
		tmp = sqrt(fabs((((2.0 * n) * U) * t)));
	} else if ((l_m <= 2e+144) || !(l_m <= 3.25e+214)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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 <= 8d-180) then
        tmp = sqrt(abs((((2.0d0 * n) * u) * t)))
    else if ((l_m <= 2d+144) .or. (.not. (l_m <= 3.25d+214))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l_m * l_m) * (((-2.0d0) + (u_42 * (n / om))) / om))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l_m * (l_m / 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 <= 8e-180) {
		tmp = Math.sqrt(Math.abs((((2.0 * n) * U) * t)));
	} else if ((l_m <= 2e+144) || !(l_m <= 3.25e+214)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 8e-180:
		tmp = math.sqrt(math.fabs((((2.0 * n) * U) * t)))
	elif (l_m <= 2e+144) or not (l_m <= 3.25e+214):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8e-180)
		tmp = sqrt(abs(Float64(Float64(Float64(2.0 * n) * U) * t)));
	elseif ((l_m <= 2e+144) || !(l_m <= 3.25e+214))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(-2.0 + Float64(U_42_ * Float64(n / Om))) / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / 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 <= 8e-180)
		tmp = sqrt(abs((((2.0 * n) * U) * t)));
	elseif ((l_m <= 2e+144) || ~((l_m <= 3.25e+214)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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, 8e-180], N[Sqrt[N[Abs[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l$95$m, 2e+144], N[Not[LessEqual[l$95$m, 3.25e+214]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.0000000000000002e-180

   1. Initial program 57.0%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in l around 0 42.7%

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

      1. associate-*r* 42.7%

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

   7. Simplified 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. pow1/2 42.7%

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

      3. pow1/2 43.4%

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

      4. pow-prod-down 30.5%

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

      5. pow2 30.5%

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

   9. Applied egg-rr 30.5%

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

       1. unpow1/2 30.5%

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

       2. unpow2 30.5%

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

       3. rem-sqrt-square 43.9%

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

       4. associate-*r* 48.5%

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

       5. associate-*r* 48.5%

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

       6. *-commutative 48.5%

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

       7. associate-*r* 48.5%

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

   11. Simplified 48.5%

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

   if 8.0000000000000002e-180 < l < 2.00000000000000005e144 or 3.25e214 < l

   1. Initial program 44.5%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in U around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. associate-/l* 44.0%

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

      3. unpow2 44.0%

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

      4. unpow2 44.0%

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

      5. times-frac 48.0%

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

      6. unpow2 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in Om around -inf 48.2%

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

      1. mul-1-neg 48.2%

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

      2. fma-define 48.2%

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

      3. associate-/l* 50.6%

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

   10. Simplified 50.6%

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

   11. Taylor expanded in l around 0 49.6%

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

       1. associate-/l* 49.6%

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

       2. sub-neg 49.6%

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

       3. associate-/l* 55.8%

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

       4. metadata-eval 55.8%

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

   13. Simplified 55.8%

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

       1. unpow2 55.8%

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

   15. Applied egg-rr 55.8%

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

   if 2.00000000000000005e144 < l < 3.25e214

   1. Initial program 50.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in Om around inf 50.8%

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

      1. unpow2 50.8%

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

      2. associate-*r/ 83.1%

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

      3. *-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+144} \lor \neg \left(\ell \leq 3.25 \cdot 10^{+214}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+144} \lor \neg \left(l\_m \leq 1.95 \cdot 10^{+214}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8e-179)
   (sqrt (* (* (* 2.0 n) U) t))
   (if (or (<= l_m 2e+144) (not (<= l_m 1.95e+214)))
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (* l_m l_m) (/ (+ -2.0 (* U* (/ n Om))) Om))))))
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l_m (/ l_m 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 <= 8e-179) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else if ((l_m <= 2e+144) || !(l_m <= 1.95e+214)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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 <= 8d-179) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else if ((l_m <= 2d+144) .or. (.not. (l_m <= 1.95d+214))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l_m * l_m) * (((-2.0d0) + (u_42 * (n / om))) / om))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l_m * (l_m / 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 <= 8e-179) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else if ((l_m <= 2e+144) || !(l_m <= 1.95e+214)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 8e-179:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	elif (l_m <= 2e+144) or not (l_m <= 1.95e+214):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8e-179)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	elseif ((l_m <= 2e+144) || !(l_m <= 1.95e+214))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(-2.0 + Float64(U_42_ * Float64(n / Om))) / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / 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 <= 8e-179)
		tmp = sqrt((((2.0 * n) * U) * t));
	elseif ((l_m <= 2e+144) || ~((l_m <= 1.95e+214)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((l_m * l_m) * ((-2.0 + (U_42_ * (n / Om))) / Om))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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, 8e-179], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l$95$m, 2e+144], N[Not[LessEqual[l$95$m, 1.95e+214]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.0000000000000002e-179

   1. Initial program 57.0%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in l around 0 42.7%

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

      1. associate-*r* 42.7%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in U around 0 42.7%

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

      1. associate-*r* 42.7%

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

      2. associate-*r* 47.3%

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

      3. associate-*r* 47.3%

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

      4. *-commutative 47.3%

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

      5. associate-*r* 47.3%

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

   10. Simplified 47.3%

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

   if 8.0000000000000002e-179 < l < 2.00000000000000005e144 or 1.95000000000000007e214 < l

   1. Initial program 44.5%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in U around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. associate-/l* 44.0%

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

      3. unpow2 44.0%

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

      4. unpow2 44.0%

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

      5. times-frac 48.0%

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

      6. unpow2 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in Om around -inf 48.2%

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

      1. mul-1-neg 48.2%

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

      2. fma-define 48.2%

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

      3. associate-/l* 50.6%

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

   10. Simplified 50.6%

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

   11. Taylor expanded in l around 0 49.6%

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

       1. associate-/l* 49.6%

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

       2. sub-neg 49.6%

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

       3. associate-/l* 55.8%

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

       4. metadata-eval 55.8%

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

   13. Simplified 55.8%

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

       1. unpow2 55.8%

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

   15. Applied egg-rr 55.8%

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

   if 2.00000000000000005e144 < l < 1.95000000000000007e214

   1. Initial program 50.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in Om around inf 50.8%

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

      1. unpow2 50.8%

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

      2. associate-*r/ 83.1%

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

      3. *-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+144} \lor \neg \left(\ell \leq 1.95 \cdot 10^{+214}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \frac{-2 + U* \cdot \frac{n}{Om}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;l\_m \leq 10^{+58}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.4e-195)
   (sqrt (* (* (* 2.0 n) U) t))
   (if (<= l_m 1e+58)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (* U* (* (* l_m l_m) (/ n Om))) Om)))))
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l_m (/ l_m 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-195) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else if (l_m <= 1e+58) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ * ((l_m * l_m) * (n / Om))) / Om)))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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-195) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else if (l_m <= 1d+58) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((u_42 * ((l_m * l_m) * (n / om))) / om)))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l_m * (l_m / 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-195) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else if (l_m <= 1e+58) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((U_42_ * ((l_m * l_m) * (n / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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-195:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	elif l_m <= 1e+58:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((U_42_ * ((l_m * l_m) * (n / Om))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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-195)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	elseif (l_m <= 1e+58)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * Float64(n / Om))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / 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-195)
		tmp = sqrt((((2.0 * n) * U) * t));
	elseif (l_m <= 1e+58)
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ * ((l_m * l_m) * (n / Om))) / Om)))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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-195], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1e+58], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.4e-195

   1. Initial program 56.5%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in l around 0 42.6%

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

      1. associate-*r* 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in U around 0 42.6%

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

      1. associate-*r* 42.6%

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

      2. associate-*r* 46.7%

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

      3. associate-*r* 46.7%

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

      4. *-commutative 46.7%

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

      5. associate-*r* 46.7%

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

   10. Simplified 46.7%

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

   if 5.4e-195 < l < 9.99999999999999944e57

   1. Initial program 50.5%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in U around 0 54.8%

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

      1. mul-1-neg 54.8%

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

      2. associate-/l* 52.5%

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

      3. unpow2 52.5%

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

      4. unpow2 52.5%

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

      5. times-frac 52.5%

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

      6. unpow2 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in Om around -inf 52.8%

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

      1. mul-1-neg 52.8%

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

      2. fma-define 52.8%

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

      3. associate-/l* 55.0%

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

   10. Simplified 55.0%

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

   11. Taylor expanded in U* around inf 49.8%

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

       1. associate-/l* 52.0%

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

       2. associate-*r/ 54.0%

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

   13. Simplified 54.0%

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

       1. unpow2 55.3%

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

   15. Applied egg-rr 54.0%

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

   if 9.99999999999999944e57 < l

   1. Initial program 40.8%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in Om around inf 43.1%

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

      1. unpow2 43.1%

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

      2. associate-*r/ 52.5%

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

      3. *-commutative 52.5%

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

   7. Applied egg-rr 52.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;\ell \leq 10^{+58}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.2 \cdot 10^{-44}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.2e-44)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l_m (/ l_m 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 <= 4.2e-44) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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 <= 4.2d-44) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l_m * (l_m / 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 <= 4.2e-44) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 4.2e-44:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / Om)))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4.2e-44)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / 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 <= 4.2e-44)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l_m * (l_m / 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, 4.2e-44], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.20000000000000003e-44

   1. Initial program 55.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in l around 0 41.3%

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

      1. pow1/2 42.3%

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

      2. associate-*r* 46.0%

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

   7. Applied egg-rr 46.0%

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

   if 4.20000000000000003e-44 < l

   1. Initial program 43.0%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in Om around inf 42.8%

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

      1. unpow2 42.8%

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

      2. associate-*r/ 49.5%

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

      3. *-commutative 49.5%

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

   7. Applied egg-rr 49.5%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-44}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9 \cdot 10^{+55}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 9e+55)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* -4.0 (/ (* U (* n (* l_m l_m))) 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 <= 9e+55) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((-4.0 * ((U * (n * (l_m * l_m))) / 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 <= 9d+55) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * ((u * (n * (l_m * l_m))) / 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 <= 9e+55) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * ((U * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 9e+55:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * ((U * (n * (l_m * l_m))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 9e+55)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(n * Float64(l_m * l_m))) / 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 <= 9e+55)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * ((U * (n * (l_m * l_m))) / 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, 9e+55], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.99999999999999996e55

   1. Initial program 55.1%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in l around 0 41.1%

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

      1. pow1/2 42.0%

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

      2. associate-*r* 45.0%

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

   7. Applied egg-rr 45.0%

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

   if 8.99999999999999996e55 < l

   1. Initial program 40.8%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in Om around inf 43.1%

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

   6. Taylor expanded in t around 0 40.5%

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

      1. unpow2 57.9%

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

   8. Applied egg-rr 40.5%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{+55}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+14}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3e+14)
   (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 <= 3e+14) {
		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 <= 3d+14) 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 <= 3e+14) {
		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 <= 3e+14:
		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 <= 3e+14)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * 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 <= 3e+14)
		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, 3e+14], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3e14

   1. Initial program 55.2%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in l around 0 41.6%

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

      1. pow1/2 42.6%

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

      2. associate-*r* 45.7%

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

   7. Applied egg-rr 45.7%

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

   if 3e14 < l

   1. Initial program 42.9%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in l around 0 19.4%

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

      1. associate-*r* 19.4%

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

   7. Simplified 19.4%

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

      1. pow1/2 23.4%

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

   9. Applied egg-rr 23.4%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+14}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.3e+68)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* (* 2.0 U) (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.3e+68) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.3d+68) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.3e+68) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.3e+68:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.3e+68)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.3e+68)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt(((2.0 * U) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.3e+68], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.3e68

   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 55.2%

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

   5. Taylor expanded in l around 0 39.6%

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

      1. associate-*r* 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in U around 0 39.6%

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

      1. associate-*r* 39.6%

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

      2. associate-*r* 43.1%

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

      3. associate-*r* 43.1%

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

      4. *-commutative 43.1%

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

      5. associate-*r* 43.2%

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

   10. Simplified 43.2%

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

   if 2.3e68 < l

   1. Initial program 46.1%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in l around 0 20.1%

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

      1. associate-*r* 20.1%

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

   7. Simplified 20.1%

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

Alternative 15: 35.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.05e+68)
   (sqrt (* t (* 2.0 (* n U))))
   (sqrt (* (* 2.0 U) (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+68) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.05d+68) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+68) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.05e+68:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.05e+68)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.05e+68)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = sqrt(((2.0 * U) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.05e+68], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.05e68

   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 57.9%

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

   5. Taylor expanded in t around inf 43.1%

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

   if 2.05e68 < l

   1. Initial program 46.1%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in l around 0 20.1%

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

      1. associate-*r* 20.1%

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

   7. Simplified 20.1%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U* \leq 3.4 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U* 3.4e-243)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* (* 2.0 U) (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 3.4e-243) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 3.4d-243) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 3.4e-243) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U_42_ <= 3.4e-243:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 3.4e-243)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 3.4e-243)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt(((2.0 * U) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U$42$, 3.4e-243], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 3.39999999999999996e-243

   1. Initial program 57.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in l around 0 40.5%

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

   if 3.39999999999999996e-243 < U*

   1. Initial program 46.9%

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

   3. Simplified 46.7%

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

   5. Taylor expanded in l around 0 37.5%

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

      1. associate-*r* 37.5%

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

   7. Simplified 37.5%

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

Alternative 17: 37.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return pow((2.0 * (t * (n * U))), 0.5);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

3. Simplified 55.4%

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

5. Taylor expanded in l around 0 36.6%

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

   1. pow1/2 37.8%

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

   2. associate-*r* 39.9%

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

7. Applied egg-rr 39.9%

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

8. Final simplification 39.9%

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

Alternative 18: 34.8% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 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(2.0 * Float64(n * Float64(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[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.8%

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

3. Simplified 55.4%

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

5. Taylor expanded in l around 0 36.6%

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

Reproduce

herbie shell --seed 2024146 
(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*))))))