Toniolo and Linder, Equation (13)

Percentage Accurate: 50.1% → 66.6%

Time: 32.1s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.1% 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: 66.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 10^{-252}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(t_1, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U))))))
   (if (<= t_2 1e-252)
     (sqrt (* 2.0 (* n (* U (fma t_1 (- U* U) (fma -2.0 (* l (/ l Om)) t))))))
     (if (<= t_2 5e+291)
       (sqrt t_2)
       (*
        (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -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 = n * pow((l / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_2 <= 1e-252) {
		tmp = sqrt((2.0 * (n * (U * fma(t_1, (U_42_ - U), fma(-2.0, (l * (l / Om)), t))))));
	} else if (t_2 <= 5e+291) {
		tmp = sqrt(t_2);
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-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(n * (Float64(l / Om) ^ 2.0))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 1e-252)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * fma(t_1, Float64(U_42_ - U), fma(-2.0, Float64(l * Float64(l / Om)), t))))));
	elseif (t_2 <= 5e+291)
		tmp = sqrt(t_2);
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + 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[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-252], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+291], N[Sqrt[t$95$2], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $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 (*.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*)))) < 9.99999999999999943e-253

   1. Initial program 18.7%

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

   3. Simplified 25.8%

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

      1. pow1 25.8%

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

   5. Applied egg-rr 25.8%

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

      1. unpow1 25.8%

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

      2. associate-*l* 43.1%

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

      3. fma-def 43.1%

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

      4. associate-*r* 45.5%

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

      5. fma-udef 45.5%

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

      6. *-commutative 45.5%

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

      7. associate-*r/ 38.1%

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

      8. unpow2 38.1%

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

      9. +-commutative 38.1%

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

      10. fma-def 38.1%

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

      11. +-commutative 38.1%

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

      12. unpow2 38.1%

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

      13. associate-*r/ 45.5%

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

      14. fma-def 45.5%

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

   7. Simplified 45.5%

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

   if 9.99999999999999943e-253 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 5.0000000000000001e291

   1. Initial program 99.7%

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

   if 5.0000000000000001e291 < (*.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 21.4%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in l around inf 21.9%

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

      1. sub-neg 21.9%

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

      2. unpow2 21.9%

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

      3. times-frac 25.0%

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

      4. associate-*r/ 25.0%

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

      5. metadata-eval 25.0%

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

      6. distribute-neg-frac 25.0%

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

      7. metadata-eval 25.0%

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

   6. Simplified 25.0%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-252}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_1 1e+146)
       t_1
       (*
        (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_1 <= 1e+146) {
		tmp = t_1;
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	}
	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 * l) / om))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else if (t_1 <= 1d+146) then
        tmp = t_1
    else
        tmp = sqrt((u * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om))))) * (l * sqrt(2.0d0))
    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 * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else if (t_1 <= 1e+146) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-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 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	elif t_1 <= 1e+146:
		tmp = t_1
	else:
		tmp = math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-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 = 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_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_1 <= 1e+146)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((2.0 * U) * (n * t)));
	elseif (t_1 <= 1e+146)
		tmp = t_1;
	else
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-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[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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+146], t$95$1, N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $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 14.4%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in t around inf 35.3%

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

      1. associate-*r* 35.5%

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

   6. Simplified 35.5%

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

   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*))))) < 9.99999999999999934e145

   1. Initial program 99.6%

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

   if 9.99999999999999934e145 < (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 20.6%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in l around inf 21.0%

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

      1. sub-neg 21.0%

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

      2. unpow2 21.0%

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

      3. times-frac 25.6%

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

      4. associate-*r/ 25.6%

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

      5. metadata-eval 25.6%

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

      6. distribute-neg-frac 25.6%

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

      7. metadata-eval 25.6%

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

   6. Simplified 25.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 3: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.8e+148)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ (+ t (* (/ (* l l) Om) -2.0)) (* n (* (pow (/ l Om) 2.0) (- U* U))))))
   (*
    (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -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 tmp;
	if (l <= 1.8e+148) {
		tmp = sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	}
	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 <= 1.8d+148) then
        tmp = sqrt(((2.0d0 * (n * u)) * ((t + (((l * l) / om) * (-2.0d0))) + (n * (((l / om) ** 2.0d0) * (u_42 - u))))))
    else
        tmp = sqrt((u * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om))))) * (l * sqrt(2.0d0))
    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 <= 1.8e+148) {
		tmp = Math.sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U))))));
	} else {
		tmp = Math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.8e+148:
		tmp = math.sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U))))))
	else:
		tmp = math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * math.sqrt(2.0))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.8e+148)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + 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_)
	tmp = 0.0;
	if (l <= 1.8e+148)
		tmp = sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (((l / Om) ^ 2.0) * (U_42_ - U))))));
	else
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-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$42$_] := If[LessEqual[l, 1.8e+148], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $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 2 regimes

2. if l < 1.80000000000000003e148

   1. Initial program 54.7%

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

   3. Simplified 54.3%

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

   if 1.80000000000000003e148 < l

   1. Initial program 14.9%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in l around inf 50.6%

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

      1. sub-neg 50.6%

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

      2. unpow2 50.6%

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

      3. times-frac 66.8%

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

      4. associate-*r/ 66.8%

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

      5. metadata-eval 66.8%

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

      6. distribute-neg-frac 66.8%

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

      7. metadata-eval 66.8%

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

   6. Simplified 66.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 4: 58.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-81}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n}{\frac{Om \cdot Om}{U - U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.25e-81)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (if (<= l 2.6e+149)
     (sqrt
      (*
       (* 2.0 (* n U))
       (- t (* (* l l) (+ (/ 2.0 Om) (/ n (/ (* Om Om) (- U U*))))))))
     (*
      (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -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 tmp;
	if (l <= 1.25e-81) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2.6e+149) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	}
	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 <= 1.25d-81) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 2.6d+149) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l * l) * ((2.0d0 / om) + (n / ((om * om) / (u - u_42))))))))
    else
        tmp = sqrt((u * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om))))) * (l * sqrt(2.0d0))
    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 <= 1.25e-81) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2.6e+149) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	} else {
		tmp = Math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.25e-81:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 2.6e+149:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))))
	else:
		tmp = math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * math.sqrt(2.0))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.25e-81)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 2.6e+149)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l * l) * Float64(Float64(2.0 / Om) + Float64(n / Float64(Float64(Om * Om) / Float64(U - U_42_))))))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + 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_)
	tmp = 0.0;
	if (l <= 1.25e-81)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 2.6e+149)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	else
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-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$42$_] := If[LessEqual[l, 1.25e-81], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.6e+149], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] + N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $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 l < 1.24999999999999995e-81

   1. Initial program 52.6%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around inf 41.6%

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

      1. associate-*r* 41.6%

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

   6. Simplified 41.6%

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

      1. pow1/2 43.9%

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

      2. associate-*l* 43.9%

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

   8. Applied egg-rr 43.9%

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

   if 1.24999999999999995e-81 < l < 2.59999999999999979e149

   1. Initial program 62.6%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in l around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*r/ 62.5%

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

      3. metadata-eval 62.5%

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

      4. associate-/l* 62.5%

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

      5. unpow2 62.5%

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

   6. Simplified 62.5%

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

   if 2.59999999999999979e149 < l

   1. Initial program 14.9%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in l around inf 50.6%

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

      1. sub-neg 50.6%

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

      2. unpow2 50.6%

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

      3. times-frac 66.8%

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

      4. associate-*r/ 66.8%

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

      5. metadata-eval 66.8%

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

      6. distribute-neg-frac 66.8%

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

      7. metadata-eval 66.8%

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

   6. Simplified 66.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-81}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n}{\frac{Om \cdot Om}{U - U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 5: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-83}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n}{\frac{Om \cdot Om}{U - U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3e-83)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (if (<= l 1.22e+151)
     (sqrt
      (*
       (* 2.0 (* n U))
       (- t (* (* l l) (+ (/ 2.0 Om) (/ n (/ (* Om Om) (- U U*))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt (* U (* n (- (/ (* n U*) (* Om Om)) (/ 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 <= 3e-83) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 1.22e+151) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * (((n * U_42_) / (Om * Om)) - (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 <= 3d-83) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 1.22d+151) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l * l) * ((2.0d0 / om) + (n / ((om * om) / (u - u_42))))))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((u * (n * (((n * u_42) / (om * om)) - (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 <= 3e-83) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 1.22e+151) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((U * (n * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3e-83:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 1.22e+151:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((U * (n * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3e-83)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 1.22e+151)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l * l) * Float64(Float64(2.0 / Om) + Float64(n / Float64(Float64(Om * Om) / Float64(U - U_42_))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / Float64(Om * Om)) - Float64(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 <= 3e-83)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 1.22e+151)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * (((n * U_42_) / (Om * Om)) - (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, 3e-83], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.22e+151], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] + N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.0000000000000001e-83

   1. Initial program 52.6%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around inf 41.6%

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

      1. associate-*r* 41.6%

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

   6. Simplified 41.6%

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

      1. pow1/2 43.9%

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

      2. associate-*l* 43.9%

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

   8. Applied egg-rr 43.9%

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

   if 3.0000000000000001e-83 < l < 1.22000000000000005e151

   1. Initial program 62.6%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in l around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*r/ 62.5%

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

      3. metadata-eval 62.5%

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

      4. associate-/l* 62.5%

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

      5. unpow2 62.5%

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

   6. Simplified 62.5%

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

   if 1.22000000000000005e151 < l

   1. Initial program 14.9%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in l around inf 50.6%

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

      1. unpow2 50.6%

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

      2. associate-*r/ 50.6%

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

      3. metadata-eval 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in U* around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. unpow2 50.6%

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

   9. Simplified 50.6%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-83}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n}{\frac{Om \cdot Om}{U - U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 6: 54.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n}{\frac{Om \cdot Om}{U - U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.5e-83)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (if (<= l 3.8e+112)
     (sqrt
      (*
       (* 2.0 (* n U))
       (- t (* (* l l) (+ (/ 2.0 Om) (/ n (/ (* Om Om) (- U U*))))))))
     (pow (* 2.0 (* (* n U) (- t (* (/ l Om) (* 2.0 l))))) 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 <= 1.5e-83) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 3.8e+112) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	} else {
		tmp = pow((2.0 * ((n * U) * (t - ((l / Om) * (2.0 * l))))), 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 <= 1.5d-83) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 3.8d+112) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l * l) * ((2.0d0 / om) + (n / ((om * om) / (u - u_42))))))))
    else
        tmp = (2.0d0 * ((n * u) * (t - ((l / om) * (2.0d0 * l))))) ** 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 <= 1.5e-83) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 3.8e+112) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	} else {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((l / Om) * (2.0 * l))))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.5e-83:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 3.8e+112:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))))
	else:
		tmp = math.pow((2.0 * ((n * U) * (t - ((l / Om) * (2.0 * l))))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.5e-83)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 3.8e+112)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l * l) * Float64(Float64(2.0 / Om) + Float64(n / Float64(Float64(Om * Om) / Float64(U - U_42_))))))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))))) ^ 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 <= 1.5e-83)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 3.8e+112)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * ((2.0 / Om) + (n / ((Om * Om) / (U - U_42_))))))));
	else
		tmp = (2.0 * ((n * U) * (t - ((l / Om) * (2.0 * l))))) ^ 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, 1.5e-83], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 3.8e+112], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] + N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.50000000000000005e-83

   1. Initial program 52.6%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around inf 41.6%

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

      1. associate-*r* 41.6%

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

   6. Simplified 41.6%

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

      1. pow1/2 43.9%

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

      2. associate-*l* 43.9%

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

   8. Applied egg-rr 43.9%

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

   if 1.50000000000000005e-83 < l < 3.80000000000000008e112

   1. Initial program 60.9%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in l around 0 60.8%

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

      1. unpow2 60.8%

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

      2. associate-*r/ 60.8%

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

      3. metadata-eval 60.8%

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

      4. associate-/l* 60.8%

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

      5. unpow2 60.8%

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

   6. Simplified 60.8%

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

   if 3.80000000000000008e112 < l

   1. Initial program 24.1%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in Om around inf 25.0%

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

      1. unpow2 25.0%

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

      2. associate-*r/ 29.6%

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

      3. associate-*r* 29.6%

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

   6. Simplified 29.6%

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

      1. pow1/2 37.5%

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

      2. associate-*l* 37.5%

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

   8. Applied egg-rr 37.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{n}{\frac{Om \cdot Om}{U - U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7: 46.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-77}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4e-77)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e-77) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	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-77) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l / om) * (2.0d0 * l)))))
    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-77) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4e-77:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4e-77)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	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-77)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	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-77], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.9999999999999997e-77

   1. Initial program 53.2%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 41.1%

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

      1. associate-*r* 41.1%

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

   6. Simplified 41.1%

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

      1. pow1/2 43.4%

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

      2. associate-*l* 43.4%

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

   8. Applied egg-rr 43.4%

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

   if 3.9999999999999997e-77 < l

   1. Initial program 40.2%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in Om around inf 32.4%

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

      1. unpow2 32.4%

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

      2. associate-*r/ 34.9%

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

      3. associate-*r* 34.9%

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

   6. Simplified 34.9%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-77}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 8: 52.5% accurate, 1.9× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (pow (* 2.0 (* (* n U) (- t (* (/ l Om) (* 2.0 l))))) 0.5))

C

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

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 = (2.0d0 * ((n * u) * (t - ((l / om) * (2.0d0 * l))))) ** 0.5d0
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.pow((2.0 * ((n * U) * (t - ((l / Om) * (2.0 * l))))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Simplified 52.0%

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

4. Taylor expanded in Om around inf 42.3%

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

   1. unpow2 42.3%

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

   2. associate-*r/ 46.1%

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

   3. associate-*r* 46.1%

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

6. Simplified 46.1%

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

   1. pow1/2 50.2%

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

   2. associate-*l* 50.2%

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

8. Applied egg-rr 50.2%

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

9. Final simplification 50.2%

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

Alternative 9: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(2 \cdot \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.8e+99)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* (* U -2.0) (* 2.0 (/ (* l (* n l)) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.8e+99) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt(((U * -2.0) * (2.0 * ((l * (n * 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 (l <= 5.8d+99) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((u * (-2.0d0)) * (2.0d0 * ((l * (n * 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 (l <= 5.8e+99) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt(((U * -2.0) * (2.0 * ((l * (n * l)) / Om))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.8e+99:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt(((U * -2.0) * (2.0 * ((l * (n * l)) / Om))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.8e+99)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(2.0 * Float64(Float64(l * Float64(n * 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 (l <= 5.8e+99)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt(((U * -2.0) * (2.0 * ((l * (n * 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[l, 5.8e+99], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(2.0 * N[(N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.8000000000000004e99

   1. Initial program 54.2%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 39.7%

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

      1. associate-*r* 41.6%

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

      2. *-commutative 41.6%

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

   6. Simplified 41.6%

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

      1. pow1/2 44.0%

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

      2. associate-*l* 44.8%

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

   8. Applied egg-rr 44.8%

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

   if 5.8000000000000004e99 < l

   1. Initial program 25.3%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in l around inf 30.8%

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

      1. associate-*r* 30.8%

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

      2. associate-*r* 33.1%

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

      3. *-commutative 33.1%

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

      4. unpow2 33.1%

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

      5. associate-*r/ 33.1%

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

      6. metadata-eval 33.1%

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

      7. unpow2 33.1%

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

      8. times-frac 37.8%

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

   6. Simplified 37.8%

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

   7. Taylor expanded in n around 0 30.2%

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

      1. *-commutative 30.2%

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

      2. unpow2 30.2%

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

      3. associate-*r* 40.8%

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

   9. Simplified 40.8%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(2 \cdot \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}\right)}\\ \end{array} \]

Alternative 10: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+101}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U}{\frac{Om}{\ell \cdot \left(n \cdot \ell\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.1e+101)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* -4.0 (/ U (/ Om (* l (* n l))))))))

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+101) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((-4.0 * (U / (Om / (l * (n * l))))));
	}
	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+101) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * (u / (om / (l * (n * l))))))
    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+101) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * (U / (Om / (l * (n * l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.1e+101:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * (U / (Om / (l * (n * l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.1e+101)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(U / Float64(Om / Float64(l * Float64(n * l))))));
	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+101)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * (U / (Om / (l * (n * l))))));
	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+101], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(U / N[(Om / N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1e101

   1. Initial program 54.2%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 39.7%

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

      1. associate-*r* 41.6%

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

      2. *-commutative 41.6%

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

   6. Simplified 41.6%

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

      1. pow1/2 44.0%

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

      2. associate-*l* 44.8%

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

   8. Applied egg-rr 44.8%

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

   if 2.1e101 < l

   1. Initial program 25.3%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in Om around inf 26.1%

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

      1. unpow2 26.1%

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

      2. associate-*r/ 30.5%

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

      3. associate-*r* 30.5%

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

   6. Simplified 30.5%

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

   7. Taylor expanded in t around 0 28.0%

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

      1. associate-/l* 30.1%

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

      2. *-commutative 30.1%

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

      3. unpow2 30.1%

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

      4. associate-*r* 40.8%

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

   9. Simplified 40.8%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+101}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U}{\frac{Om}{\ell \cdot \left(n \cdot \ell\right)}}}\\ \end{array} \]

Alternative 11: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq -1.4 \cdot 10^{+73}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -1.4e+73)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* 2.0 (* n (* U t))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -1.4e+73) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (n * (U * 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 (u_42 <= (-1.4d+73)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (n * (u * 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 (U_42_ <= -1.4e+73) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -1.4e+73:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -1.4e+73)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -1.4e+73)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (n * (U * 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[U$42$, -1.4e+73], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -1.40000000000000004e73

   1. Initial program 48.2%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in t around inf 27.7%

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

      1. associate-*r* 27.7%

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

   6. Simplified 27.7%

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

      1. pow1/2 37.0%

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

      2. associate-*l* 37.0%

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

   8. Applied egg-rr 37.0%

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

   if -1.40000000000000004e73 < U*

   1. Initial program 49.4%

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

   3. Simplified 53.4%

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

      1. pow1 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. unpow1 53.4%

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

      2. associate-*l* 53.6%

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

      3. fma-def 53.6%

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

      4. associate-*r* 55.7%

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

      5. fma-udef 55.7%

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

      6. *-commutative 55.7%

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

      7. associate-*r/ 51.1%

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

      8. unpow2 51.1%

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

      9. +-commutative 51.1%

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

      10. fma-def 51.1%

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

      11. +-commutative 51.1%

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

      12. unpow2 51.1%

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

      13. associate-*r/ 55.7%

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

      14. fma-def 55.7%

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

   7. Simplified 55.7%

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

   8. Taylor expanded in l around 0 39.9%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -1.4 \cdot 10^{+73}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 12: 37.0% accurate, 2.1× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* n (* U t))) 0.5))

C

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

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 = (2.0d0 * (n * (u * t))) ** 0.5d0
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.pow((2.0 * (n * (U * t))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Simplified 54.0%

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

4. Taylor expanded in l around 0 33.9%

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

   1. associate-*r* 35.5%

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

   2. *-commutative 35.5%

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

6. Simplified 35.5%

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

   1. pow1/2 37.9%

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

   2. associate-*l* 38.6%

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

8. Applied egg-rr 38.6%

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

9. Final simplification 38.6%

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

Alternative 13: 35.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (n * (U * 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[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Simplified 54.0%

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

   1. pow1 54.0%

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

5. Applied egg-rr 54.0%

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

   1. unpow1 54.0%

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

   2. associate-*l* 53.7%

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

   3. fma-def 53.7%

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

   4. associate-*r* 55.3%

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

   5. fma-udef 55.3%

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

   6. *-commutative 55.3%

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

   7. associate-*r/ 49.9%

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

   8. unpow2 49.9%

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

   9. +-commutative 49.9%

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

   10. fma-def 49.9%

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

   11. +-commutative 49.9%

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

   12. unpow2 49.9%

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

   13. associate-*r/ 55.3%

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

   14. fma-def 55.3%

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

7. Simplified 55.3%

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

8. Taylor expanded in l around 0 36.2%

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

9. Final simplification 36.2%

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

Reproduce

herbie shell --seed 2023279 
(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*))))))