Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 64.4%

Time: 25.8s

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: 49.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(n \cdot t_1\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(\frac{1}{\frac{Om}{{\ell}^{2}}}, -2, t\right) + t_1 \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2 (* (* n t_1) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt
       (*
        U
        (+ (fma (/ 1.0 (/ Om (pow l 2.0))) -2.0 t) (* t_1 (* n (- U* U)))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 (* l (/ l Om)))) t_2)))
       (*
        (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 2.0 Om)))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = (n * t_1) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (fma((1.0 / (Om / pow(l, 2.0))), -2.0, t) + (t_1 * (n * (U_42_ - U))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * (l * (l / Om)))) + t_2)));
	} else {
		tmp = sqrt((U * (n * ((n / (pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(Float64(n * t_1) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(fma(Float64(1.0 / Float64(Om / (l ^ 2.0))), -2.0, t) + Float64(t_1 * Float64(n * Float64(U_42_ - U)))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_2)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(n / Float64((Om ^ 2.0) / Float64(U_42_ - U))) - Float64(2.0 / Om))))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(N[(N[(1.0 / N[(Om / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] + N[(t$95$1 * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * 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 18.3%

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

   3. Simplified 18.3%

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

   5. Applied egg-rr 35.0%

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

      1. *-commutative 35.0%

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

      2. *-commutative 35.0%

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

   7. Simplified 35.0%

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

      1. clear-num 35.1%

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

      2. inv-pow 35.1%

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

   9. Applied egg-rr 35.1%

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

       1. unpow-1 35.1%

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

   11. Simplified 35.1%

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

   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 66.7%

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

      1. associate-*l/ 70.3%

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

   3. Applied egg-rr 70.3%

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

   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 11.3%

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

   4. Taylor expanded in l around inf 17.4%

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

      1. associate-/l* 17.5%

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

      2. associate-*r/ 17.5%

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

      3. metadata-eval 17.5%

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

   6. Simplified 17.5%

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

4. Final simplification 58.3%

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

Alternative 2: 64.1% accurate, 0.3× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 t_1)))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (*
        (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 2.0 Om)))))
        (* l (sqrt 2.0)))))))

C

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

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * t_1))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.sqrt((U * (n * ((n / (Math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * t_1))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.sqrt((U * (n * ((n / (math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l * math.sqrt(2.0))
	return tmp

Julia

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

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = sqrt((U * (n * ((n / ((Om ^ 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * 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 18.3%

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

   3. Simplified 18.3%

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

   5. Applied egg-rr 35.0%

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

      1. *-commutative 35.0%

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

      2. *-commutative 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in n around 0 35.0%

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

      1. unpow2 35.0%

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

      2. associate-*l/ 35.0%

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

   10. Applied egg-rr 35.0%

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

   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 66.7%

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

      1. associate-*l/ 70.3%

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

   3. Applied egg-rr 70.3%

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

   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 11.3%

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

   4. Taylor expanded in l around inf 17.4%

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

      1. associate-/l* 17.5%

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

      2. associate-*r/ 17.5%

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

      3. metadata-eval 17.5%

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

   6. Simplified 17.5%

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

4. Final simplification 58.3%

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

Alternative 3: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot t_1\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 t_1)))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (pow (* (* 2.0 n) (* U (fma -2.0 (/ (pow l 2.0) Om) t))) 0.5)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = pow(((2.0 * n) * (U * fma(-2.0, (pow(l, 2.0) / Om), t))), 0.5);
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * t_1)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * fma(-2.0, Float64((l ^ 2.0) / Om), t))) ^ 0.5;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] + t), $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 18.3%

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

   3. Simplified 18.3%

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

   5. Applied egg-rr 35.0%

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

      1. *-commutative 35.0%

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

      2. *-commutative 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in n around 0 35.0%

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

      1. unpow2 35.0%

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

      2. associate-*l/ 35.0%

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

   10. Applied egg-rr 35.0%

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

   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 66.7%

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

      1. associate-*l/ 70.3%

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

   3. Applied egg-rr 70.3%

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

   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 0.5%

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

   5. Applied egg-rr 0.5%

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

      1. *-commutative 0.5%

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

      2. *-commutative 0.5%

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

   7. Simplified 0.5%

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

   8. Taylor expanded in n around 0 6.6%

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

      1. *-commutative 6.6%

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

      2. pow1/2 26.0%

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

      3. pow1/2 26.0%

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

      4. *-commutative 26.0%

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

      5. pow-prod-down 35.4%

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

      6. +-commutative 35.4%

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

      7. fma-def 35.4%

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

   10. Applied egg-rr 35.4%

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

4. Final simplification 60.8%

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

Alternative 4: 57.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot t_1\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_4 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 t_1)))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (sqrt
        (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * t_1))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * t_1))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * t_1)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 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 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 17.7%

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

   3. Simplified 17.7%

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

   5. Applied egg-rr 33.9%

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

      1. *-commutative 33.9%

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

      2. *-commutative 33.9%

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

   7. Simplified 33.9%

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

   8. Taylor expanded in n around 0 34.0%

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

      1. unpow2 34.0%

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

      2. associate-*l/ 34.0%

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

   10. Applied egg-rr 34.0%

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

   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 66.7%

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

      1. associate-*l/ 70.3%

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

   3. Applied egg-rr 70.3%

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

   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%

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

   2. Taylor expanded in Om around inf 20.6%

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

4. Final simplification 58.6%

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

Alternative 5: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -1.85 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 n)
           (*
            U
            (+
             (+ t (* -2.0 (/ l (/ Om l))))
             (* n (* (pow (/ l Om) 2.0) (- U* U)))))))))
   (if (<= n -1.85e-122)
     t_1
     (if (<= n 1.05e-276)
       (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
       (if (<= n 5.5e-121)
         (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 (* l (/ l Om)))))))
         t_1)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (pow((l / Om), 2.0) * (U_42_ - U)))))));
	double tmp;
	if (n <= -1.85e-122) {
		tmp = t_1;
	} else if (n <= 1.05e-276) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (n <= 5.5e-121) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * (u * ((t + ((-2.0d0) * (l / (om / l)))) + (n * (((l / om) ** 2.0d0) * (u_42 - u)))))))
    if (n <= (-1.85d-122)) then
        tmp = t_1
    else if (n <= 1.05d-276) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else if (n <= 5.5d-121) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * (l * (l / om))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U)))))));
	double tmp;
	if (n <= -1.85e-122) {
		tmp = t_1;
	} else if (n <= 1.05e-276) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (n <= 5.5e-121) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U)))))))
	tmp = 0
	if n <= -1.85e-122:
		tmp = t_1
	elif n <= 1.05e-276:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	elif n <= 5.5e-121:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * (l * (l / Om))))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U)))))))
	tmp = 0.0
	if (n <= -1.85e-122)
		tmp = t_1;
	elseif (n <= 1.05e-276)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (n <= 5.5e-121)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (((l / Om) ^ 2.0) * (U_42_ - U)))))));
	tmp = 0.0;
	if (n <= -1.85e-122)
		tmp = t_1;
	elseif (n <= 1.05e-276)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	elseif (n <= 5.5e-121)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.85e-122], t$95$1, If[LessEqual[n, 1.05e-276], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 5.5e-121], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.8499999999999999e-122 or 5.50000000000000031e-121 < n

   1. Initial program 54.3%

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

   3. Simplified 58.6%

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

   if -1.8499999999999999e-122 < n < 1.05e-276

   1. Initial program 55.8%

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

   2. Taylor expanded in Om around inf 60.5%

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

   if 1.05e-276 < n < 5.50000000000000031e-121

   1. Initial program 32.0%

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

   3. Simplified 39.4%

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

   5. Applied egg-rr 47.6%

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

      1. *-commutative 47.6%

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

      2. *-commutative 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in n around 0 48.4%

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

      1. unpow2 48.4%

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

      2. associate-*l/ 60.1%

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

   10. Applied egg-rr 60.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 48.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 8.8 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{1}{\frac{Om}{U \cdot {\ell}^{2}}} + U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 8.8e-281)
   (sqrt (* 2.0 (* n (+ (* -2.0 (/ 1.0 (/ Om (* U (pow l 2.0))))) (* U t)))))
   (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 (* l (/ l Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 8.8e-281) {
		tmp = sqrt((2.0 * (n * ((-2.0 * (1.0 / (Om / (U * pow(l, 2.0))))) + (U * t)))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (n <= 8.8d-281) then
        tmp = sqrt((2.0d0 * (n * (((-2.0d0) * (1.0d0 / (om / (u * (l ** 2.0d0))))) + (u * t)))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * (l * (l / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 8.8e-281) {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * (1.0 / (Om / (U * Math.pow(l, 2.0))))) + (U * t)))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 8.8e-281:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * (1.0 / (Om / (U * math.pow(l, 2.0))))) + (U * t)))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * (l * (l / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 8.8e-281)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(1.0 / Float64(Om / Float64(U * (l ^ 2.0))))) + Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 8.8e-281)
		tmp = sqrt((2.0 * (n * ((-2.0 * (1.0 / (Om / (U * (l ^ 2.0))))) + (U * t)))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 8.8e-281], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(1.0 / N[(Om / N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.80000000000000016e-281

   1. Initial program 56.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in Om around inf 50.6%

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

      1. clear-num 50.6%

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

      2. inv-pow 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. unpow-1 50.6%

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

   8. Simplified 50.6%

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

   if 8.80000000000000016e-281 < n

   1. Initial program 45.4%

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

   3. Simplified 46.9%

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

   5. Applied egg-rr 51.0%

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

      1. *-commutative 51.0%

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

      2. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   8. Taylor expanded in n around 0 46.1%

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

      1. unpow2 46.1%

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

      2. associate-*l/ 51.6%

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

   10. Applied egg-rr 51.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.8 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{1}{\frac{Om}{U \cdot {\ell}^{2}}} + U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 7: 48.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 5.6e-275)
   (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
   (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 (* l (/ l Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 5.6e-275) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (n <= 5.6d-275) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * (l * (l / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 5.6e-275) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 5.6e-275:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * (l * (l / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 5.6e-275)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 5.6e-275)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 5.6e-275], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.59999999999999989e-275

   1. Initial program 56.2%

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

   2. Taylor expanded in Om around inf 51.4%

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

   if 5.59999999999999989e-275 < n

   1. Initial program 45.4%

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

   3. Simplified 46.9%

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

   5. Applied egg-rr 51.0%

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

      1. *-commutative 51.0%

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

      2. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   8. Taylor expanded in n around 0 46.1%

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

      1. unpow2 46.1%

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

      2. associate-*l/ 51.6%

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

   10. Applied egg-rr 51.6%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 8: 48.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 7.2e-281)
   (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (pow l 2.0)) Om))))))
   (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 (* l (/ l Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 7.2e-281) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * pow(l, 2.0)) / Om))))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (n <= 7.2d-281) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * ((u * (l ** 2.0d0)) / om))))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * (l * (l / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 7.2e-281) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * Math.pow(l, 2.0)) / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 7.2e-281:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * math.pow(l, 2.0)) / Om))))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * (l * (l / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 7.2e-281)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) / Om))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 7.2e-281)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l ^ 2.0)) / Om))))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 7.2e-281], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 7.20000000000000013e-281

   1. Initial program 56.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in Om around inf 50.6%

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

   if 7.20000000000000013e-281 < n

   1. Initial program 45.4%

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

   3. Simplified 46.9%

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

   5. Applied egg-rr 51.0%

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

      1. *-commutative 51.0%

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

      2. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   8. Taylor expanded in n around 0 46.1%

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

      1. unpow2 46.1%

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

      2. associate-*l/ 51.6%

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

   10. Applied egg-rr 51.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 9: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 3.1 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 3.1e-308)
   (sqrt (* 2.0 (* (- t (/ 2.0 (/ Om (pow l 2.0)))) (* n U))))
   (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 (* l (/ l Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 3.1e-308) {
		tmp = sqrt((2.0 * ((t - (2.0 / (Om / pow(l, 2.0)))) * (n * U))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (n <= 3.1d-308) then
        tmp = sqrt((2.0d0 * ((t - (2.0d0 / (om / (l ** 2.0d0)))) * (n * u))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * (l * (l / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 3.1e-308) {
		tmp = Math.sqrt((2.0 * ((t - (2.0 / (Om / Math.pow(l, 2.0)))) * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 3.1e-308:
		tmp = math.sqrt((2.0 * ((t - (2.0 / (Om / math.pow(l, 2.0)))) * (n * U))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * (l * (l / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 3.1e-308)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t - Float64(2.0 / Float64(Om / (l ^ 2.0)))) * Float64(n * U))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 3.1e-308)
		tmp = sqrt((2.0 * ((t - (2.0 / (Om / (l ^ 2.0)))) * (n * U))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * (l * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 3.1e-308], N[Sqrt[N[(2.0 * N[(N[(t - N[(2.0 / N[(Om / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.09999999999999992e-308

   1. Initial program 56.4%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in Om around inf 49.1%

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

      1. associate-*r/ 49.1%

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

      2. associate-/l* 49.1%

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

   6. Simplified 49.1%

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

   if 3.09999999999999992e-308 < n

   1. Initial program 45.7%

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

   3. Simplified 47.0%

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

   5. Applied egg-rr 50.7%

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

      1. *-commutative 50.7%

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

      2. *-commutative 50.7%

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

   7. Simplified 50.7%

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

   8. Taylor expanded in n around 0 46.7%

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

      1. unpow2 46.7%

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

      2. associate-*l/ 52.0%

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

   10. Applied egg-rr 52.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.1 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - \frac{2}{\frac{Om}{{\ell}^{2}}}\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 10: 40.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+55}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(U \cdot \left(n \cdot {\ell}^{2}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4e-171)
   (sqrt (* 2.0 (* n (* U t))))
   (if (<= l 1.4e+55)
     (pow (* (* n t) (* 2.0 U)) 0.5)
     (sqrt (* 2.0 (/ (* -2.0 (* U (* n (pow l 2.0)))) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e-171) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else if (l <= 1.4e+55) {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = sqrt((2.0 * ((-2.0 * (U * (n * pow(l, 2.0)))) / Om)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (l <= 4d-171) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else if (l <= 1.4d+55) then
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (((-2.0d0) * (u * (n * (l ** 2.0d0)))) / om)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e-171) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else if (l <= 1.4e+55) {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * ((-2.0 * (U * (n * Math.pow(l, 2.0)))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4e-171:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	elif l <= 1.4e+55:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	else:
		tmp = math.sqrt((2.0 * ((-2.0 * (U * (n * math.pow(l, 2.0)))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4e-171)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	elseif (l <= 1.4e+55)
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(-2.0 * Float64(U * Float64(n * (l ^ 2.0)))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4e-171)
		tmp = sqrt((2.0 * (n * (U * t))));
	elseif (l <= 1.4e+55)
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	else
		tmp = sqrt((2.0 * ((-2.0 * (U * (n * (l ^ 2.0)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4e-171], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.4e+55], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(-2.0 * N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.9999999999999999e-171

   1. Initial program 51.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 40.6%

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

   if 3.9999999999999999e-171 < l < 1.4e55

   1. Initial program 63.6%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in t around inf 42.4%

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

      1. pow1/2 44.9%

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

      2. associate-*r* 44.9%

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

   6. Applied egg-rr 44.9%

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

   if 1.4e55 < l

   1. Initial program 35.2%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in Om around inf 33.1%

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

   5. Taylor expanded in l around inf 30.2%

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

      1. associate-*r/ 30.2%

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

      2. *-commutative 30.2%

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

   7. Simplified 30.2%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+55}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(U \cdot \left(n \cdot {\ell}^{2}\right)\right)}{Om}}\\ \end{array} \]

Alternative 11: 44.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.4e-173)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.4e-173) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (l <= 5.4d-173) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.4e-173) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.4e-173:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.4e-173)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.4e-173)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.4e-173], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.3999999999999999e-173

   1. Initial program 51.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 40.6%

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

   if 5.3999999999999999e-173 < l

   1. Initial program 49.5%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in n around 0 41.0%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 38.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-189}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -1.8e-189)
   (pow (* (* n t) (* 2.0 U)) 0.5)
   (sqrt (fabs (* t (* 2.0 (* n U)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.8e-189) {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = sqrt(fabs((t * (2.0 * (n * U)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (t <= (-1.8d-189)) then
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    else
        tmp = sqrt(abs((t * (2.0d0 * (n * u)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.8e-189) {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = Math.sqrt(Math.abs((t * (2.0 * (n * U)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -1.8e-189:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	else:
		tmp = math.sqrt(math.fabs((t * (2.0 * (n * U)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -1.8e-189)
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = sqrt(abs(Float64(t * Float64(2.0 * Float64(n * U)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.8e-189)
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	else
		tmp = sqrt(abs((t * (2.0 * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -1.8e-189], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.80000000000000008e-189

   1. Initial program 51.6%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around inf 46.4%

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

      1. pow1/2 47.4%

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

      2. associate-*r* 47.4%

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

   6. Applied egg-rr 47.4%

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

   if -1.80000000000000008e-189 < t

   1. Initial program 50.7%

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

      1. associate-*l/ 53.7%

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

   3. Applied egg-rr 53.7%

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

   4. Taylor expanded in t around inf 31.8%

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

      1. add-sqr-sqrt 31.8%

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

      2. pow1/2 31.8%

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

      3. pow1/2 33.1%

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

      4. pow-prod-down 24.3%

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

      5. pow2 24.3%

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

      6. *-commutative 24.3%

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

      7. associate-*l* 24.3%

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

   6. Applied egg-rr 24.3%

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

      1. unpow1/2 24.3%

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

      2. unpow2 24.3%

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

      3. rem-sqrt-square 34.0%

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

      4. *-commutative 34.0%

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

   8. Simplified 34.0%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-189}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \]

Alternative 13: 37.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3e-170)
   (sqrt (* 2.0 (* n (* U t))))
   (pow (* (* n t) (* 2.0 U)) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e-170) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (l <= 3d-170) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e-170) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3e-170:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3e-170)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3e-170)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3e-170], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.00000000000000013e-170

   1. Initial program 51.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 40.6%

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

   if 3.00000000000000013e-170 < l

   1. Initial program 49.5%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in t around inf 26.4%

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

      1. pow1/2 27.7%

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

      2. associate-*r* 27.7%

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

   6. Applied egg-rr 27.7%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \]

Alternative 14: 35.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.1e-169)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* U (* n t))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.1e-169) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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
    real(8) :: tmp
    if (l <= 2.1d-169) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.1e-169) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.1e-169:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.1e-169)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.1e-169)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.1e-169], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1000000000000001e-169

   1. Initial program 51.7%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 40.6%

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

   if 2.1000000000000001e-169 < l

   1. Initial program 49.5%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in t around inf 26.4%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15: 35.2% accurate, 2.1× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

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

Fortran

NOTE: l should be positive before calling this function
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 * (u * (n * t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Simplified 51.5%

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

4. Taylor expanded in t around inf 34.4%

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

5. Final simplification 34.4%

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

Reproduce

herbie shell --seed 2023320 
(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*))))))