Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 64.9%

Time: 28.1s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l_m}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{l_m}^{2}}{Om}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot U*}{{Om}^{2}} - \frac{2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(l_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U)))))))
   (if (<= t_2 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (fma 2.0 (/ (pow l_m 2.0) Om) (* n (* t_1 (- U U*))))))))
     (if (<= t_2 5e+147)
       t_2
       (*
        (sqrt (* (- (/ (* n U*) (pow Om 2.0)) (/ 2.0 Om)) (* n U)))
        (* 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, (pow(l_m, 2.0) / Om), (n * (t_1 * (U - U_42_)))))));
	} else if (t_2 <= 5e+147) {
		tmp = t_2;
	} else {
		tmp = sqrt(((((n * U_42_) / pow(Om, 2.0)) - (2.0 / Om)) * (n * U))) * (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 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64((l_m ^ 2.0) / Om), Float64(n * Float64(t_1 * Float64(U - U_42_))))))));
	elseif (t_2 <= 5e+147)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(n * U_42_) / (Om ^ 2.0)) - Float64(2.0 / Om)) * Float64(n * U))) * 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[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 * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $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[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+147], t$95$2, N[(N[Sqrt[N[(N[(N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.6%

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

   3. Simplified 46.5%

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

      1. sqrt-prod 61.3%

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

      2. fma-udef 61.3%

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

      3. associate-*l/ 61.3%

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

      4. associate-*r/ 61.3%

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

      5. associate--l- 61.3%

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

      6. associate--l- 61.3%

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

   5. Applied egg-rr 61.3%

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

      1. *-commutative 61.3%

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

      2. *-commutative 61.3%

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

   7. Simplified 61.3%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.0000000000000002e147

   1. Initial program 97.9%

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

   if 5.0000000000000002e147 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 24.7%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in l around inf 27.2%

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

   5. Taylor expanded in U around 0 27.7%

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

      1. associate-*r* 28.4%

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

      2. *-commutative 28.4%

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

      3. *-commutative 28.4%

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

      4. associate-*r/ 28.4%

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

      5. metadata-eval 28.4%

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

   7. Simplified 28.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot U*}{{Om}^{2}} - \frac{2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2: 62.4% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (* n (pow (/ l_m Om) 2.0)))
        (t_3
         (sqrt (* t_1 (- (- t (* 2.0 (/ (* l_m l_m) Om))) (* t_2 (- U U*)))))))
   (if (<= t_3 0.0)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) (* t_2 (- U* U)))))
       (pow
        (fma -4.0 (/ U (/ Om (* n (pow l_m 2.0)))) (* 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_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = n * pow((l_m / Om), 2.0);
	double t_3 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - (t_2 * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l_m * (l_m / Om)))) + (t_2 * (U_42_ - U)))));
	} else {
		tmp = pow(fma(-4.0, (U / (Om / (n * pow(l_m, 2.0)))), (2.0 * (t * (n * U)))), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	t_3 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(t_2 * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + Float64(t_2 * Float64(U_42_ - U)))));
	else
		tmp = fma(-4.0, Float64(U / Float64(Om / Float64(n * (l_m ^ 2.0)))), Float64(2.0 * Float64(t * Float64(n * U)))) ^ 0.5;
	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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.6%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in Om around inf 49.4%

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

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

   1. Initial program 69.0%

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

      1. associate-*l/ 73.1%

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

   3. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 9.5%

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

   4. Taylor expanded in Om around inf 10.5%

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

      1. pow1/2 33.7%

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

      2. fma-def 33.7%

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

      3. associate-/l* 35.7%

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

      4. *-commutative 35.7%

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

      5. associate-*r* 36.0%

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

   6. Applied egg-rr 36.0%

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

4. Final simplification 63.4%

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

Alternative 3: 65.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.5%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in Om around inf 48.1%

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

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

   1. Initial program 69.0%

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

      1. associate-*l/ 73.1%

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

   3. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in l around inf 26.0%

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

      1. sqrt-prod 5.3%

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

      2. div-inv 5.3%

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

      3. pow-flip 5.3%

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

      4. metadata-eval 5.3%

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

      5. un-div-inv 5.3%

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

   6. Applied egg-rr 5.3%

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

      1. associate-*l* 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in U around 0 5.3%

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

       1. associate-/l* 5.4%

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

       2. associate-*r/ 5.4%

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

       3. metadata-eval 5.4%

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

   11. Simplified 5.4%

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

       1. expm1-log1p-u 5.4%

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

       2. expm1-udef 0.7%

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

       3. sqrt-unprod 16.7%

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

       4. associate-/r/ 16.6%

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

   13. Applied egg-rr 16.6%

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

       1. expm1-def 26.2%

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

       2. expm1-log1p 26.2%

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

       3. *-commutative 26.2%

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

       4. associate-*l* 26.1%

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

       5. associate-*l/ 26.0%

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

       6. *-commutative 26.0%

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

   15. Simplified 26.0%

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

4. Final simplification 61.9%

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

Alternative 4: 64.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot U*}{{Om}^{2}} - \frac{2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(l_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 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) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_1 2e+295)
       (sqrt t_1)
       (*
        (sqrt (* (- (/ (* n U*) (pow Om 2.0)) (/ 2.0 Om)) (* n U)))
        (* 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 = ((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) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (t_1 <= 2e+295) {
		tmp = sqrt(t_1);
	} else {
		tmp = sqrt(((((n * U_42_) / pow(Om, 2.0)) - (2.0 / Om)) * (n * U))) * (l_m * sqrt(2.0));
	}
	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 = ((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) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else if (t_1 <= 2d+295) then
        tmp = sqrt(t_1)
    else
        tmp = sqrt(((((n * u_42) / (om ** 2.0d0)) - (2.0d0 / om)) * (n * u))) * (l_m * sqrt(2.0d0))
    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 * 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) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (t_1 <= 2e+295) {
		tmp = Math.sqrt(t_1);
	} else {
		tmp = Math.sqrt(((((n * U_42_) / Math.pow(Om, 2.0)) - (2.0 / Om)) * (n * U))) * (l_m * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = ((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) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif t_1 <= 2e+295:
		tmp = math.sqrt(t_1)
	else:
		tmp = math.sqrt(((((n * U_42_) / math.pow(Om, 2.0)) - (2.0 / Om)) * (n * U))) * (l_m * math.sqrt(2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(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 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_1 <= 2e+295)
		tmp = sqrt(t_1);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(n * U_42_) / (Om ^ 2.0)) - Float64(2.0 / Om)) * Float64(n * U))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = ((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) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (t_1 <= 2e+295)
		tmp = sqrt(t_1);
	else
		tmp = sqrt(((((n * U_42_) / (Om ^ 2.0)) - (2.0 / Om)) * (n * U))) * (l_m * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(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]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+295], N[Sqrt[t$95$1], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.5%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in Om around inf 48.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 2e295

   1. Initial program 97.9%

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

   if 2e295 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 25.7%

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

   3. Simplified 33.9%

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

   4. Taylor expanded in l around inf 27.9%

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

   5. Taylor expanded in U around 0 28.5%

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

      1. associate-*r* 29.2%

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

      2. *-commutative 29.2%

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

      3. *-commutative 29.2%

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

      4. associate-*r/ 29.2%

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

      5. metadata-eval 29.2%

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

   7. Simplified 29.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot U*}{{Om}^{2}} - \frac{2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 5: 61.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.5%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in Om around inf 48.1%

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

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

   1. Initial program 69.0%

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

      1. associate-*l/ 73.1%

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

   3. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in U* around inf 34.2%

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

4. Final simplification 63.2%

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

Alternative 6: 60.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.5%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in Om around inf 48.1%

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

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

   1. Initial program 69.0%

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

      1. associate-*l/ 73.1%

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

   3. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in U* around inf 34.2%

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

   5. Taylor expanded in n around -inf 33.5%

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

      1. mul-1-neg 33.5%

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

      2. *-commutative 33.5%

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

      3. associate-*r* 33.5%

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

      4. distribute-rgt-neg-in 33.5%

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

      5. associate-*r* 33.5%

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

      6. associate-/l* 33.4%

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

   7. Simplified 33.4%

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

4. Final simplification 63.1%

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

Alternative 7: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-56} \lor \neg \left(n \leq 2.4 \cdot 10^{-100}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + n \cdot \left({\left(\frac{l_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {l_m}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -5.5e-56) (not (<= n 2.4e-100)))
   (sqrt
    (*
     (* (* 2.0 n) U)
     (+
      (- t (* 2.0 (* l_m (/ l_m Om))))
      (* n (* (pow (/ l_m Om) 2.0) (- U* U))))))
   (sqrt (+ (* -4.0 (/ (* U (* n (pow l_m 2.0))) Om)) (* 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 ((n <= -5.5e-56) || !(n <= 2.4e-100)) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * (l_m * (l_m / Om)))) + (n * (pow((l_m / Om), 2.0) * (U_42_ - U))))));
	} else {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l_m, 2.0))) / Om)) + (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 ((n <= (-5.5d-56)) .or. (.not. (n <= 2.4d-100))) then
        tmp = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * (l_m * (l_m / om)))) + (n * (((l_m / om) ** 2.0d0) * (u_42 - u))))))
    else
        tmp = sqrt((((-4.0d0) * ((u * (n * (l_m ** 2.0d0))) / om)) + (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 ((n <= -5.5e-56) || !(n <= 2.4e-100)) {
		tmp = 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))))));
	} else {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l_m, 2.0))) / Om)) + (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 (n <= -5.5e-56) or not (n <= 2.4e-100):
		tmp = 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))))))
	else:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l_m, 2.0))) / Om)) + (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 ((n <= -5.5e-56) || !(n <= 2.4e-100))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * Float64(U_42_ - U))))));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l_m ^ 2.0))) / Om)) + Float64(2.0 * Float64(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 ((n <= -5.5e-56) || ~((n <= 2.4e-100)))
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * (l_m * (l_m / Om)))) + (n * (((l_m / Om) ^ 2.0) * (U_42_ - U))))));
	else
		tmp = sqrt(((-4.0 * ((U * (n * (l_m ^ 2.0))) / Om)) + (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[Or[LessEqual[n, -5.5e-56], N[Not[LessEqual[n, 2.4e-100]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.4999999999999999e-56 or 2.4000000000000003e-100 < n

   1. Initial program 52.0%

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

      1. associate-*l/ 57.3%

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

   3. Applied egg-rr 57.3%

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

      1. associate-*r* 58.6%

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

      2. expm1-log1p-u 47.1%

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

      3. expm1-udef 45.9%

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

   5. Applied egg-rr 45.9%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 58.6%

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

      3. *-commutative 58.6%

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

   7. Simplified 58.6%

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

   if -5.4999999999999999e-56 < n < 2.4000000000000003e-100

   1. Initial program 43.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in Om around inf 58.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-56} \lor \neg \left(n \leq 2.4 \cdot 10^{-100}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 8: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\\ \mathbf{if}\;l_m \leq 1.95 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(l_m \cdot l_m\right)}{Om}\right) + n \cdot \left({\left(\frac{l_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;l_m \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {l_m}^{2}\right)}{Om} + t_1}\\ \mathbf{elif}\;l_m \leq 2.9 \cdot 10^{+169}:\\ \;\;\;\;\sqrt[3]{{t_1}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* U (* n t)))))
   (if (<= l_m 1.95e+98)
     (sqrt
      (*
       (* 2.0 n)
       (*
        U
        (+
         (- t (/ (* 2.0 (* l_m l_m)) Om))
         (* n (* (pow (/ l_m Om) 2.0) (- U* U)))))))
     (if (<= l_m 3.4e+146)
       (sqrt (+ (* -4.0 (/ (* U (* n (pow l_m 2.0))) Om)) t_1))
       (if (<= l_m 2.9e+169)
         (cbrt (pow t_1 1.5))
         (* (* l_m (sqrt 2.0)) (sqrt (* -2.0 (/ (* n U) Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (U * (n * t));
	double tmp;
	if (l_m <= 1.95e+98) {
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	} else if (l_m <= 3.4e+146) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l_m, 2.0))) / Om)) + t_1));
	} else if (l_m <= 2.9e+169) {
		tmp = cbrt(pow(t_1, 1.5));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * ((n * U) / Om)));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (U * (n * t));
	double tmp;
	if (l_m <= 1.95e+98) {
		tmp = 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)))))));
	} else if (l_m <= 3.4e+146) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l_m, 2.0))) / Om)) + t_1));
	} else if (l_m <= 2.9e+169) {
		tmp = Math.cbrt(Math.pow(t_1, 1.5));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((-2.0 * ((n * U) / Om)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(U * Float64(n * t)))
	tmp = 0.0
	if (l_m <= 1.95e+98)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om)) + Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	elseif (l_m <= 3.4e+146)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l_m ^ 2.0))) / Om)) + t_1));
	elseif (l_m <= 2.9e+169)
		tmp = cbrt((t_1 ^ 1.5));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(-2.0 * Float64(Float64(n * U) / Om))));
	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[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 1.95e+98], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.4e+146], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.9e+169], N[Power[N[Power[t$95$1, 1.5], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.95e98

   1. Initial program 50.9%

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

   3. Simplified 51.8%

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

   if 1.95e98 < l < 3.39999999999999991e146

   1. Initial program 48.4%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in Om around inf 81.7%

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

   if 3.39999999999999991e146 < l < 2.9000000000000001e169

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in l around 0 39.5%

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

      1. add-cbrt-cube 51.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. pow1/3 51.3%

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

      3. add-sqr-sqrt 51.3%

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

      4. pow1 51.3%

         \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 63.8%

         \[\leadsto {\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 63.8%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-*r* 51.3%

         \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 51.3%

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

   6. Applied egg-rr 51.3%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 51.3%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{1.5}}} \]

      2. associate-*r* 63.8%

         \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)}^{1.5}} \]

   8. Simplified 63.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1.5}}} \]

   if 2.9000000000000001e169 < 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 41.4%

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

   4. Taylor expanded in l around inf 73.3%

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

   5. Taylor expanded in n around 0 44.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+169}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \]

Alternative 9: 48.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 6.5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{l_m}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.5e+157)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l_m 2.0) Om)))))))
   (* (* l_m (sqrt 2.0)) (sqrt (* -2.0 (/ (* n U) 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 <= 6.5e+157) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * ((n * U) / 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 <= 6.5d+157) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((-2.0d0) * ((n * u) / 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 <= 6.5e+157) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((-2.0 * ((n * U) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 6.5e+157:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((-2.0 * ((n * U) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.5e+157)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(-2.0 * Float64(Float64(n * U) / 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 <= 6.5e+157)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l_m ^ 2.0) / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * ((n * U) / 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, 6.5e+157], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.5e157

   1. Initial program 51.2%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in n around 0 48.0%

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

   if 6.5e157 < l

   1. Initial program 26.2%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in l around inf 75.5%

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

   5. Taylor expanded in n around 0 45.0%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \]

Alternative 10: 42.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(n \cdot \left(2 \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.2e+46)
   (sqrt (fabs (* U (* n (* 2.0 t)))))
   (* (* l_m (sqrt 2.0)) (sqrt (* -2.0 (/ U (/ Om n)))))))

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+46) {
		tmp = sqrt(fabs((U * (n * (2.0 * t)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * (U / (Om / n))));
	}
	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+46) then
        tmp = sqrt(abs((u * (n * (2.0d0 * t)))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((-2.0d0) * (u / (om / n))))
    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+46) {
		tmp = Math.sqrt(Math.abs((U * (n * (2.0 * t)))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((-2.0 * (U / (Om / n))));
	}
	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+46:
		tmp = math.sqrt(math.fabs((U * (n * (2.0 * t)))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((-2.0 * (U / (Om / n))))
	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+46)
		tmp = sqrt(abs(Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(-2.0 * Float64(U / Float64(Om / n)))));
	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+46)
		tmp = sqrt(abs((U * (n * (2.0 * t)))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * (U / (Om / n))));
	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+46], N[Sqrt[N[Abs[N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2e46

   1. Initial program 50.8%

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

      1. associate-*l/ 53.0%

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

   3. Applied egg-rr 53.0%

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

      1. associate-*r* 53.5%

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

      2. expm1-log1p-u 45.1%

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

      3. expm1-udef 44.2%

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

   5. Applied egg-rr 44.2%

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

      1. expm1-def 45.1%

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

      2. expm1-log1p 53.5%

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

      3. *-commutative 53.5%

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

   7. Simplified 53.5%

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

   8. Taylor expanded in t around inf 41.7%

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

      1. associate-*r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-*r* 37.1%

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

      4. associate-*r* 37.1%

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

      5. *-commutative 37.1%

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

   10. Simplified 37.1%

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

       1. *-commutative 37.1%

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

       2. associate-*r* 37.1%

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

       3. pow1 37.1%

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

       4. metadata-eval 37.1%

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

       5. metadata-eval 37.1%

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

       6. metadata-eval 37.1%

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

       7. pow-prod-up 38.1%

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

       8. pow-prod-down 30.1%

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

       9. pow2 30.1%

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

       10. associate-*r* 30.1%

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

       11. *-commutative 30.1%

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

       12. associate-*l* 30.1%

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

       13. metadata-eval 30.1%

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

   12. Applied egg-rr 30.1%

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

       1. unpow1/2 30.1%

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

       2. unpow2 30.1%

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

       3. rem-sqrt-square 38.7%

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

       4. *-commutative 38.7%

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

       5. associate-*l* 38.7%

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

       6. associate-*r* 43.4%

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

       7. *-commutative 43.4%

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

       8. *-commutative 43.4%

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

       9. associate-*l* 43.4%

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

       10. associate-*l* 43.4%

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

   14. Simplified 43.4%

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

   if 4.2e46 < 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 47.8%

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

   4. Taylor expanded in l around inf 61.9%

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

   5. Taylor expanded in n around 0 32.6%

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

      1. associate-/l* 38.2%

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

   7. Simplified 38.2%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(n \cdot \left(2 \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\\ \end{array} \]

Alternative 11: 39.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(n \cdot \left(2 \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 1.8e-66)
   (sqrt (fabs (* U (* n (* 2.0 t)))))
   (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_) {
	double tmp;
	if (U <= 1.8e-66) {
		tmp = sqrt(fabs((U * (n * (2.0 * t)))));
	} else {
		tmp = pow((2.0 * (t * (n * U))), 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 (u <= 1.8d-66) then
        tmp = sqrt(abs((u * (n * (2.0d0 * t)))))
    else
        tmp = (2.0d0 * (t * (n * u))) ** 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 (U <= 1.8e-66) {
		tmp = Math.sqrt(Math.abs((U * (n * (2.0 * t)))));
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 1.8e-66:
		tmp = math.sqrt(math.fabs((U * (n * (2.0 * t)))))
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 1.8e-66)
		tmp = sqrt(abs(Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 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 (U <= 1.8e-66)
		tmp = sqrt(abs((U * (n * (2.0 * t)))));
	else
		tmp = (2.0 * (t * (n * U))) ^ 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[U, 1.8e-66], N[Sqrt[N[Abs[N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.80000000000000006e-66

   1. Initial program 44.4%

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

      1. associate-*l/ 48.7%

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

   3. Applied egg-rr 48.7%

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

      1. associate-*r* 49.1%

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

      2. expm1-log1p-u 40.2%

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

      3. expm1-udef 39.3%

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

   5. Applied egg-rr 39.3%

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

      1. expm1-def 40.2%

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

      2. expm1-log1p 49.1%

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

      3. *-commutative 49.1%

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

   7. Simplified 49.1%

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

   8. Taylor expanded in t around inf 36.0%

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

      1. associate-*r* 29.2%

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

      2. *-commutative 29.2%

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

      3. associate-*r* 31.6%

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

      4. associate-*r* 31.6%

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

      5. *-commutative 31.6%

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

   10. Simplified 31.6%

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

       1. *-commutative 31.6%

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

       2. associate-*r* 31.6%

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

       3. pow1 31.6%

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

       4. metadata-eval 31.6%

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

       5. metadata-eval 31.6%

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

       6. metadata-eval 31.6%

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

       7. pow-prod-up 32.7%

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

       8. pow-prod-down 26.4%

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

       9. pow2 26.4%

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

       10. associate-*r* 26.4%

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

       11. *-commutative 26.4%

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

       12. associate-*l* 26.4%

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

       13. metadata-eval 26.4%

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

   12. Applied egg-rr 26.4%

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

       1. unpow1/2 26.4%

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

       2. unpow2 26.4%

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

       3. rem-sqrt-square 33.6%

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

       4. *-commutative 33.6%

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

       5. associate-*l* 33.6%

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

       6. associate-*r* 39.0%

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

       7. *-commutative 39.0%

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

       8. *-commutative 39.0%

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

       9. associate-*l* 39.0%

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

       10. associate-*l* 39.0%

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

   14. Simplified 39.0%

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

   if 1.80000000000000006e-66 < U

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in l around 0 36.2%

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

      1. pow1/2 38.0%

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

      2. associate-*r* 47.3%

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

   6. Applied egg-rr 47.3%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(n \cdot \left(2 \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 12: 40.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 1.06 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(n \cdot \left(2 \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 1.06e-253)
   (sqrt (fabs (* U (* n (* 2.0 t)))))
   (* (sqrt (* 2.0 U)) (sqrt (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 1.06e-253) {
		tmp = sqrt(fabs((U * (n * (2.0 * t)))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 1.06d-253) then
        tmp = sqrt(abs((u * (n * (2.0d0 * t)))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 1.06e-253) {
		tmp = Math.sqrt(Math.abs((U * (n * (2.0 * t)))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 1.06e-253:
		tmp = math.sqrt(math.fabs((U * (n * (2.0 * t)))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 1.06e-253)
		tmp = sqrt(abs(Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.06e-253)
		tmp = sqrt(abs((U * (n * (2.0 * t)))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, 1.06e-253], N[Sqrt[N[Abs[N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.06000000000000007e-253

   1. Initial program 44.7%

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

      1. associate-*l/ 48.8%

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

   3. Applied egg-rr 48.8%

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

      1. associate-*r* 49.2%

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

      2. expm1-log1p-u 40.4%

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

      3. expm1-udef 39.2%

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

   5. Applied egg-rr 39.2%

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

      1. expm1-def 40.4%

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

      2. expm1-log1p 49.2%

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

      3. *-commutative 49.2%

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

   7. Simplified 49.2%

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

   8. Taylor expanded in t around inf 37.5%

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

      1. associate-*r* 30.2%

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

      2. *-commutative 30.2%

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

      3. associate-*r* 32.7%

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

      4. associate-*r* 32.7%

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

      5. *-commutative 32.7%

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

   10. Simplified 32.7%

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

       1. *-commutative 32.7%

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

       2. associate-*r* 32.7%

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

       3. pow1 32.7%

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

       4. metadata-eval 32.7%

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

       5. metadata-eval 32.7%

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

       6. metadata-eval 32.7%

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

       7. pow-prod-up 33.9%

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

       8. pow-prod-down 27.7%

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

       9. pow2 27.7%

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

       10. associate-*r* 27.7%

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

       11. *-commutative 27.7%

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

       12. associate-*l* 27.7%

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

       13. metadata-eval 27.7%

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

   12. Applied egg-rr 27.7%

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

       1. unpow1/2 27.7%

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

       2. unpow2 27.7%

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

       3. rem-sqrt-square 34.9%

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

       4. *-commutative 34.9%

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

       5. associate-*l* 34.9%

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

       6. associate-*r* 41.0%

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

       7. *-commutative 41.0%

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

       8. *-commutative 41.0%

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

       9. associate-*l* 41.0%

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

       10. associate-*l* 41.0%

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

   14. Simplified 41.0%

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

   if 1.06000000000000007e-253 < U

   1. Initial program 55.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in l around 0 33.6%

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

      1. pow1/2 34.9%

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

      2. associate-*r* 34.9%

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

      3. unpow-prod-down 44.9%

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

      4. pow1/2 43.8%

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

   6. Applied egg-rr 43.8%

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

      1. unpow1/2 43.8%

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

   8. Simplified 43.8%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.06 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(n \cdot \left(2 \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 13: 37.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 5.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 5.5e+105)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 5.5e+105) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 5.5d+105) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 5.5e+105) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 5.5e+105:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 5.5e+105)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 5.5e+105)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 5.5e+105], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.49999999999999979e105

   1. Initial program 49.0%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in l around 0 39.0%

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

   if 5.49999999999999979e105 < n

   1. Initial program 48.1%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in l around 0 18.7%

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

      1. associate-*r* 26.4%

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

      2. *-commutative 26.4%

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

   6. Simplified 26.4%

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

      1. pow1/2 31.8%

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

      2. associate-*l* 32.0%

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

   8. Applied egg-rr 32.0%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 14: 37.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 5.4 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 5.4e-66)
   (sqrt (* 2.0 (* U (* n t))))
   (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_) {
	double tmp;
	if (U <= 5.4e-66) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (t * (n * U))), 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 (u <= 5.4d-66) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (t * (n * u))) ** 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 (U <= 5.4e-66) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 5.4e-66:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 5.4e-66)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 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 (U <= 5.4e-66)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (t * (n * U))) ^ 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[U, 5.4e-66], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 5.39999999999999992e-66

   1. Initial program 44.4%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in l around 0 36.0%

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

   if 5.39999999999999992e-66 < U

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in l around 0 36.2%

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

      1. pow1/2 38.0%

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

      2. associate-*r* 47.3%

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

   6. Applied egg-rr 47.3%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 5.4 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 15: 36.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 52.4%

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

4. Taylor expanded in l around 0 36.0%

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

5. Final simplification 36.0%

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

Reproduce

herbie shell --seed 2023318 
(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*))))))