Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 62.0%

Time: 31.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 62.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, l\_m \cdot \frac{l\_m}{Om}, n \cdot \left(t\_1 \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)\right)}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U)))))))
   (if (<= t_2 2e-118)
     (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
     (if (<= t_2 4e+73)
       t_2
       (if (<= t_2 INFINITY)
         (sqrt
          (*
           (* 2.0 n)
           (* U (- t (fma 2.0 (* l_m (/ l_m Om)) (* n (* t_1 (- U U*))))))))
         (*
          (*
           l_m
           (sqrt (* n (* U (fma (/ n (pow Om 2.0)) (- U* U) (/ (- 2.0) Om))))))
          (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow((l_m / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-118) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_2 <= 4e+73) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * (U * (t - fma(2.0, (l_m * (l_m / Om)), (n * (t_1 * (U - U_42_))))))));
	} else {
		tmp = (l_m * sqrt((n * (U * fma((n / pow(Om, 2.0)), (U_42_ - U), (-2.0 / Om)))))) * sqrt(2.0);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-118)
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	elseif (t_2 <= 4e+73)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, Float64(l_m * Float64(l_m / Om)), Float64(n * Float64(t_1 * Float64(U - U_42_))))))));
	else
		tmp = Float64(Float64(l_m * sqrt(Float64(n * Float64(U * fma(Float64(n / (Om ^ 2.0)), Float64(U_42_ - U), Float64(Float64(-2.0) / Om)))))) * sqrt(2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-118], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$2, 4e+73], t$95$2, If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[(n * N[(U * N[(N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[((-2.0) / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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*))))) < 1.99999999999999997e-118

   1. Initial program 17.9%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in n around 0 46.0%

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

      1. pow1/2 45.9%

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

      2. associate-*r* 46.2%

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

      3. unpow-prod-down 39.9%

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

      4. pow1/2 39.9%

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

      5. cancel-sign-sub-inv 39.9%

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

      6. metadata-eval 39.9%

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

   7. Applied egg-rr 39.9%

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

      1. unpow1/2 39.9%

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

      2. associate-*r/ 39.9%

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

   9. Simplified 39.9%

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

       1. *-commutative 39.9%

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

       2. pow1/2 39.9%

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

       3. pow1/2 39.9%

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

       4. pow-prod-down 46.2%

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

       5. associate-/l* 46.2%

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

   11. Applied egg-rr 46.2%

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

   if 1.99999999999999997e-118 < (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*))))) < 3.99999999999999993e73

   1. Initial program 96.6%

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

   if 3.99999999999999993e73 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 52.9%

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

   3. Simplified 67.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.4%

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

   5. Taylor expanded in l around inf 30.3%

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

      1. associate-/l* 30.3%

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

      2. associate-*r/ 30.3%

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

      3. metadata-eval 30.3%

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

   7. Simplified 30.3%

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

      1. add-cbrt-cube 20.5%

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

      2. pow3 20.5%

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

      3. associate-*r* 20.4%

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

      4. *-commutative 20.4%

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

   9. Applied egg-rr 20.4%

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

       1. rem-cbrt-cube 32.6%

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

       2. associate-*r* 32.6%

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

       3. associate-*l* 32.9%

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

       4. associate-/r/ 32.9%

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

       5. fma-neg 32.9%

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

   11. Applied egg-rr 32.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-118}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(\frac{n}{{Om}^{2}}, U* - U, \frac{-2}{Om}\right)\right)}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{t\_2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, l\_m \cdot \frac{l\_m}{Om}, n \cdot \left(t\_1 \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U))))))
   (if (<= t_2 2e-236)
     (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
     (if (<= t_2 5e+146)
       (sqrt t_2)
       (if (<= t_2 INFINITY)
         (sqrt
          (*
           (* 2.0 n)
           (* U (- t (fma 2.0 (* l_m (/ l_m Om)) (* n (* t_1 (- U U*))))))))
         (*
          (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 2.0 Om)))))
          (* l_m (sqrt 2.0))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 2e-236) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_2 <= 5e+146) {
		tmp = sqrt(t_2);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * (U * (t - fma(2.0, (l_m * (l_m / Om)), (n * (t_1 * (U - U_42_))))))));
	} else {
		tmp = sqrt((U * (n * ((n / (pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 2e-236)
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	elseif (t_2 <= 5e+146)
		tmp = sqrt(t_2);
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, Float64(l_m * Float64(l_m / Om)), Float64(n * Float64(t_1 * Float64(U - U_42_))))))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(n / Float64((Om ^ 2.0) / Float64(U_42_ - U))) - Float64(2.0 / Om))))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-236], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$2, 5e+146], N[Sqrt[t$95$2], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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*)))) < 2.0000000000000001e-236

   1. Initial program 17.0%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 43.7%

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

      1. pow1/2 43.7%

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

      2. associate-*r* 44.0%

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

      3. unpow-prod-down 38.0%

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

      4. pow1/2 37.9%

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

      5. cancel-sign-sub-inv 37.9%

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

      6. metadata-eval 37.9%

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

   7. Applied egg-rr 37.9%

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

      1. unpow1/2 37.9%

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

      2. associate-*r/ 37.9%

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

   9. Simplified 37.9%

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

       1. *-commutative 37.9%

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

       2. pow1/2 38.0%

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

       3. pow1/2 38.0%

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

       4. pow-prod-down 44.0%

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

       5. associate-/l* 44.0%

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

   11. Applied egg-rr 44.0%

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

   if 2.0000000000000001e-236 < (*.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*)))) < 4.9999999999999999e146

   1. Initial program 96.6%

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

   if 4.9999999999999999e146 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 52.9%

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

   3. Simplified 67.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-/l* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

   7. Simplified 32.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-118} \lor \neg \left(t\_3 \leq \infty\right):\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) + t\_1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (or (<= t_3 2e-118) (not (<= t_3 INFINITY)))
     (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
     (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if ((t_3 <= 2e-118) || !(t_3 <= ((double) INFINITY))) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if ((t_3 <= 2e-118) || !(t_3 <= Double.POSITIVE_INFINITY)) {
		tmp = Math.pow(((n * (t + (-2.0 / (Om / Math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if (t_3 <= 2e-118) or not (t_3 <= math.inf):
		tmp = math.pow(((n * (t + (-2.0 / (Om / math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5)
	else:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if ((t_3 <= 2e-118) || !(t_3 <= Inf))
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if ((t_3 <= 2e-118) || ~((t_3 <= Inf)))
		tmp = ((n * (t + (-2.0 / (Om / (l_m ^ 2.0))))) * (2.0 * U)) ^ 0.5;
	else
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$3, 2e-118], N[Not[LessEqual[t$95$3, Infinity]], $MachinePrecision]], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 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*))))) < 1.99999999999999997e-118 or +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 8.9%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in n around 0 26.6%

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

      1. pow1/2 42.7%

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

      2. associate-*r* 42.8%

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

      3. unpow-prod-down 32.4%

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

      4. pow1/2 21.7%

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

      5. cancel-sign-sub-inv 21.7%

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

      6. metadata-eval 21.7%

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

   7. Applied egg-rr 21.7%

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

      1. unpow1/2 21.7%

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

      2. associate-*r/ 21.7%

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

   9. Simplified 21.7%

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

       1. *-commutative 21.7%

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

       2. pow1/2 32.4%

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

       3. pow1/2 32.4%

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

       4. pow-prod-down 42.8%

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

       5. associate-/l* 42.9%

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

   11. Applied egg-rr 42.9%

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

   if 1.99999999999999997e-118 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 73.1%

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

      1. associate-*l/ 78.4%

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

   4. Applied egg-rr 78.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-118} \lor \neg \left(\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty\right):\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 2e-236)
     (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 2.0 Om)))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 2e-236) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = sqrt((U * (n * ((n / (pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 2e-236) {
		tmp = Math.pow(((n * (t + (-2.0 / (Om / Math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = Math.sqrt((U * (n * ((n / (Math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l_m * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_3 <= 2e-236:
		tmp = math.pow(((n * (t + (-2.0 / (Om / math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5)
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = math.sqrt((U * (n * ((n / (math.pow(Om, 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l_m * math.sqrt(2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_3 <= 2e-236)
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(n / Float64((Om ^ 2.0) / Float64(U_42_ - U))) - Float64(2.0 / Om))))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_3 <= 2e-236)
		tmp = ((n * (t + (-2.0 / (Om / (l_m ^ 2.0))))) * (2.0 * U)) ^ 0.5;
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = sqrt((U * (n * ((n / ((Om ^ 2.0) / (U_42_ - U))) - (2.0 / Om))))) * (l_m * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-236], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 2.0000000000000001e-236

   1. Initial program 17.0%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 43.7%

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

      1. pow1/2 43.7%

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

      2. associate-*r* 44.0%

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

      3. unpow-prod-down 38.0%

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

      4. pow1/2 37.9%

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

      5. cancel-sign-sub-inv 37.9%

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

      6. metadata-eval 37.9%

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

   7. Applied egg-rr 37.9%

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

      1. unpow1/2 37.9%

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

      2. associate-*r/ 37.9%

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

   9. Simplified 37.9%

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

       1. *-commutative 37.9%

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

       2. pow1/2 38.0%

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

       3. pow1/2 38.0%

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

       4. pow-prod-down 44.0%

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

       5. associate-/l* 44.0%

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

   11. Applied egg-rr 44.0%

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

   if 2.0000000000000001e-236 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 73.1%

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

      1. associate-*l/ 78.4%

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

   4. Applied egg-rr 78.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-/l* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

   7. Simplified 32.0%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(l\_m \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 2e-236)
     (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (sqrt 2.0)
        (*
         l_m
         (sqrt (* U (* n (- (/ (* n U*) (pow Om 2.0)) (/ 2.0 Om)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 2e-236) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = sqrt(2.0) * (l_m * sqrt((U * (n * (((n * U_42_) / pow(Om, 2.0)) - (2.0 / Om))))));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 2e-236) {
		tmp = Math.pow(((n * (t + (-2.0 / (Om / Math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = Math.sqrt(2.0) * (l_m * Math.sqrt((U * (n * (((n * U_42_) / Math.pow(Om, 2.0)) - (2.0 / Om))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_3 <= 2e-236:
		tmp = math.pow(((n * (t + (-2.0 / (Om / math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5)
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = math.sqrt(2.0) * (l_m * math.sqrt((U * (n * (((n * U_42_) / math.pow(Om, 2.0)) - (2.0 / Om))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_3 <= 2e-236)
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(sqrt(2.0) * Float64(l_m * sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / (Om ^ 2.0)) - Float64(2.0 / Om)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_3 <= 2e-236)
		tmp = ((n * (t + (-2.0 / (Om / (l_m ^ 2.0))))) * (2.0 * U)) ^ 0.5;
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = sqrt(2.0) * (l_m * sqrt((U * (n * (((n * U_42_) / (Om ^ 2.0)) - (2.0 / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-236], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l$95$m * N[Sqrt[N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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*)))) < 2.0000000000000001e-236

   1. Initial program 17.0%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 43.7%

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

      1. pow1/2 43.7%

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

      2. associate-*r* 44.0%

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

      3. unpow-prod-down 38.0%

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

      4. pow1/2 37.9%

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

      5. cancel-sign-sub-inv 37.9%

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

      6. metadata-eval 37.9%

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

   7. Applied egg-rr 37.9%

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

      1. unpow1/2 37.9%

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

      2. associate-*r/ 37.9%

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

   9. Simplified 37.9%

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

       1. *-commutative 37.9%

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

       2. pow1/2 38.0%

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

       3. pow1/2 38.0%

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

       4. pow-prod-down 44.0%

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

       5. associate-/l* 44.0%

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

   11. Applied egg-rr 44.0%

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

   if 2.0000000000000001e-236 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 73.1%

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

      1. associate-*l/ 78.4%

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

   4. Applied egg-rr 78.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-/l* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

   7. Simplified 32.0%

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

      1. add-cbrt-cube 21.7%

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

      2. pow3 21.7%

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

      3. associate-*r* 21.5%

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

      4. *-commutative 21.5%

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

   9. Applied egg-rr 21.5%

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

       1. rem-cbrt-cube 34.3%

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

       2. expm1-log1p-u 32.9%

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

       3. expm1-udef 32.9%

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

   11. Applied egg-rr 33.2%

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

       1. expm1-def 33.1%

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

       2. expm1-log1p 34.6%

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

       3. *-commutative 34.6%

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

       4. associate-*r* 34.7%

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

       5. *-commutative 34.7%

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

       6. *-commutative 34.7%

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

       7. associate-*r* 34.4%

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

       8. *-commutative 34.4%

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

       9. distribute-neg-frac 34.4%

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

       10. metadata-eval 34.4%

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

   13. Simplified 34.4%

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

   14. Taylor expanded in U around 0 30.0%

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

       1. *-commutative 30.0%

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

       2. associate-*r/ 30.0%

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

       3. metadata-eval 30.0%

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

   16. Simplified 30.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U*}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 2e-236)
     (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) U*)) (/ 2.0 Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 2e-236) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((n / (pow(Om, 2.0) / U_42_)) - (2.0 / Om)))));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 2e-236) {
		tmp = Math.pow(((n * (t + (-2.0 / (Om / Math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((n / (Math.pow(Om, 2.0) / U_42_)) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_3 <= 2e-236:
		tmp = math.pow(((n * (t + (-2.0 / (Om / math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5)
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * ((n / (math.pow(Om, 2.0) / U_42_)) - (2.0 / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_3 <= 2e-236)
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(n / Float64((Om ^ 2.0) / U_42_)) - Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_3 <= 2e-236)
		tmp = ((n * (t + (-2.0 / (Om / (l_m ^ 2.0))))) * (2.0 * U)) ^ 0.5;
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((n / ((Om ^ 2.0) / U_42_)) - (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-236], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 2.0000000000000001e-236

   1. Initial program 17.0%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 43.7%

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

      1. pow1/2 43.7%

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

      2. associate-*r* 44.0%

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

      3. unpow-prod-down 38.0%

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

      4. pow1/2 37.9%

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

      5. cancel-sign-sub-inv 37.9%

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

      6. metadata-eval 37.9%

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

   7. Applied egg-rr 37.9%

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

      1. unpow1/2 37.9%

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

      2. associate-*r/ 37.9%

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

   9. Simplified 37.9%

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

       1. *-commutative 37.9%

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

       2. pow1/2 38.0%

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

       3. pow1/2 38.0%

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

       4. pow-prod-down 44.0%

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

       5. associate-/l* 44.0%

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

   11. Applied egg-rr 44.0%

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

   if 2.0000000000000001e-236 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 73.1%

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

      1. associate-*l/ 78.4%

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

   4. Applied egg-rr 78.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-/l* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in U* around inf 30.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-236}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U*}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{if}\;l\_m \leq 1.72 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{l\_m}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;l\_m \leq 1.75 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;l\_m \leq 3.3 \cdot 10^{+240}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{\frac{Om}{n \cdot U}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (pow
          (pow (* 2.0 (* (* n U) (+ t (* -2.0 (* l_m (/ l_m Om)))))) 1.5)
          0.3333333333333333)))
   (if (<= l_m 1.72e-43)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= l_m 1.35e+154)
       (pow (* (* n (+ t (/ -2.0 (/ Om (pow l_m 2.0))))) (* 2.0 U)) 0.5)
       (if (<= l_m 1.75e+207)
         t_1
         (if (<= l_m 3.3e+240)
           (* (* l_m (sqrt 2.0)) (sqrt (/ -2.0 (/ Om (* n U)))))
           t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow(pow((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))), 1.5), 0.3333333333333333);
	double tmp;
	if (l_m <= 1.72e-43) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (l_m <= 1.35e+154) {
		tmp = pow(((n * (t + (-2.0 / (Om / pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (l_m <= 1.75e+207) {
		tmp = t_1;
	} else if (l_m <= 3.3e+240) {
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 / (Om / (n * U))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * ((n * u) * (t + ((-2.0d0) * (l_m * (l_m / om)))))) ** 1.5d0) ** 0.3333333333333333d0
    if (l_m <= 1.72d-43) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else if (l_m <= 1.35d+154) then
        tmp = ((n * (t + ((-2.0d0) / (om / (l_m ** 2.0d0))))) * (2.0d0 * u)) ** 0.5d0
    else if (l_m <= 1.75d+207) then
        tmp = t_1
    else if (l_m <= 3.3d+240) then
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((-2.0d0) / (om / (n * u))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.pow(Math.pow((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))), 1.5), 0.3333333333333333);
	double tmp;
	if (l_m <= 1.72e-43) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (l_m <= 1.35e+154) {
		tmp = Math.pow(((n * (t + (-2.0 / (Om / Math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5);
	} else if (l_m <= 1.75e+207) {
		tmp = t_1;
	} else if (l_m <= 3.3e+240) {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((-2.0 / (Om / (n * U))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.pow(math.pow((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))), 1.5), 0.3333333333333333)
	tmp = 0
	if l_m <= 1.72e-43:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif l_m <= 1.35e+154:
		tmp = math.pow(((n * (t + (-2.0 / (Om / math.pow(l_m, 2.0))))) * (2.0 * U)), 0.5)
	elif l_m <= 1.75e+207:
		tmp = t_1
	elif l_m <= 3.3e+240:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((-2.0 / (Om / (n * U))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = (Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 1.5) ^ 0.3333333333333333
	tmp = 0.0
	if (l_m <= 1.72e-43)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (l_m <= 1.35e+154)
		tmp = Float64(Float64(n * Float64(t + Float64(-2.0 / Float64(Om / (l_m ^ 2.0))))) * Float64(2.0 * U)) ^ 0.5;
	elseif (l_m <= 1.75e+207)
		tmp = t_1;
	elseif (l_m <= 3.3e+240)
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(-2.0 / Float64(Om / Float64(n * U)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = ((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))) ^ 1.5) ^ 0.3333333333333333;
	tmp = 0.0;
	if (l_m <= 1.72e-43)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (l_m <= 1.35e+154)
		tmp = ((n * (t + (-2.0 / (Om / (l_m ^ 2.0))))) * (2.0 * U)) ^ 0.5;
	elseif (l_m <= 1.75e+207)
		tmp = t_1;
	elseif (l_m <= 3.3e+240)
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 / (Om / (n * U))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, If[LessEqual[l$95$m, 1.72e-43], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.35e+154], N[Power[N[(N[(n * N[(t + N[(-2.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 1.75e+207], t$95$1, If[LessEqual[l$95$m, 3.3e+240], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 / N[(Om / N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.72000000000000005e-43

   1. Initial program 60.0%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in Om around inf 56.2%

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

   if 1.72000000000000005e-43 < l < 1.35000000000000003e154

   1. Initial program 59.3%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in n around 0 57.7%

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

      1. pow1/2 65.4%

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

      2. associate-*r* 65.5%

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

      3. unpow-prod-down 32.9%

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

      4. pow1/2 30.4%

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

      5. cancel-sign-sub-inv 30.4%

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

      6. metadata-eval 30.4%

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

   7. Applied egg-rr 30.4%

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

      1. unpow1/2 30.4%

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

      2. associate-*r/ 30.4%

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

   9. Simplified 30.4%

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

       1. *-commutative 30.4%

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

       2. pow1/2 32.9%

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

       3. pow1/2 32.9%

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

       4. pow-prod-down 65.5%

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

       5. associate-/l* 65.5%

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

   11. Applied egg-rr 65.5%

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

   if 1.35000000000000003e154 < l < 1.75000000000000014e207 or 3.2999999999999998e240 < l

   1. Initial program 10.6%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in n around 0 12.1%

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

      1. add-cbrt-cube 12.1%

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

      2. pow1/3 12.1%

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

   7. Applied egg-rr 35.6%

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

      1. unpow2 12.1%

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

      2. associate-*l/ 31.1%

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

   9. Applied egg-rr 48.6%

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

   if 1.75000000000000014e207 < l < 3.2999999999999998e240

   1. Initial program 1.4%

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

   3. Simplified 2.0%

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

   5. Taylor expanded in l around inf 51.8%

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

      1. associate-/l* 51.8%

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

      2. associate-*r/ 51.8%

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

      3. metadata-eval 51.8%

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

   7. Simplified 51.8%

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

   8. Taylor expanded in n around 0 38.3%

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

      1. pow1/2 38.3%

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

      2. associate-*r/ 38.3%

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

      3. *-commutative 38.3%

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

   10. Applied egg-rr 38.3%

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

       1. unpow1/2 38.3%

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

       2. associate-/l* 38.6%

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

       3. *-commutative 38.6%

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

   12. Applied egg-rr 38.6%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.72 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{\left(\left(n \cdot \left(t + \frac{-2}{\frac{Om}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+207}:\\ \;\;\;\;{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+240}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{\frac{Om}{n \cdot U}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{if}\;l\_m \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om}\right) + n \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.56 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;l\_m \leq 3.3 \cdot 10^{+240}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{\frac{Om}{n \cdot U}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (pow
          (pow (* 2.0 (* (* n U) (+ t (* -2.0 (* l_m (/ l_m Om)))))) 1.5)
          0.3333333333333333)))
   (if (<= l_m 6.8e+153)
     (sqrt
      (*
       (* 2.0 n)
       (*
        U
        (+
         (- t (/ (* 2.0 (* l_m l_m)) Om))
         (* n (* (pow (/ l_m Om) 2.0) (- U* U)))))))
     (if (<= l_m 1.56e+206)
       t_1
       (if (<= l_m 3.3e+240)
         (* (* l_m (sqrt 2.0)) (sqrt (/ -2.0 (/ Om (* n U)))))
         t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow(pow((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))), 1.5), 0.3333333333333333);
	double tmp;
	if (l_m <= 6.8e+153) {
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	} else if (l_m <= 1.56e+206) {
		tmp = t_1;
	} else if (l_m <= 3.3e+240) {
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 / (Om / (n * U))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * ((n * u) * (t + ((-2.0d0) * (l_m * (l_m / om)))))) ** 1.5d0) ** 0.3333333333333333d0
    if (l_m <= 6.8d+153) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t - ((2.0d0 * (l_m * l_m)) / om)) + (n * (((l_m / om) ** 2.0d0) * (u_42 - u)))))))
    else if (l_m <= 1.56d+206) then
        tmp = t_1
    else if (l_m <= 3.3d+240) then
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((-2.0d0) / (om / (n * u))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.pow(Math.pow((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))), 1.5), 0.3333333333333333);
	double tmp;
	if (l_m <= 6.8e+153) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (Math.pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	} else if (l_m <= 1.56e+206) {
		tmp = t_1;
	} else if (l_m <= 3.3e+240) {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((-2.0 / (Om / (n * U))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.pow(math.pow((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))), 1.5), 0.3333333333333333)
	tmp = 0
	if l_m <= 6.8e+153:
		tmp = math.sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (math.pow((l_m / Om), 2.0) * (U_42_ - U)))))))
	elif l_m <= 1.56e+206:
		tmp = t_1
	elif l_m <= 3.3e+240:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((-2.0 / (Om / (n * U))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = (Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 1.5) ^ 0.3333333333333333
	tmp = 0.0
	if (l_m <= 6.8e+153)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om)) + Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	elseif (l_m <= 1.56e+206)
		tmp = t_1;
	elseif (l_m <= 3.3e+240)
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(-2.0 / Float64(Om / Float64(n * U)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = ((2.0 * ((n * U) * (t + (-2.0 * (l_m * (l_m / Om)))))) ^ 1.5) ^ 0.3333333333333333;
	tmp = 0.0;
	if (l_m <= 6.8e+153)
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (((l_m / Om) ^ 2.0) * (U_42_ - U)))))));
	elseif (l_m <= 1.56e+206)
		tmp = t_1;
	elseif (l_m <= 3.3e+240)
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 / (Om / (n * U))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, If[LessEqual[l$95$m, 6.8e+153], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.56e+206], t$95$1, If[LessEqual[l$95$m, 3.3e+240], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 / N[(Om / N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.7999999999999995e153

   1. Initial program 59.9%

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

   3. Simplified 62.8%

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

   if 6.7999999999999995e153 < l < 1.5599999999999999e206 or 3.2999999999999998e240 < l

   1. Initial program 10.6%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in n around 0 12.1%

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

      1. add-cbrt-cube 12.1%

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

      2. pow1/3 12.1%

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

   7. Applied egg-rr 35.6%

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

      1. unpow2 12.1%

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

      2. associate-*l/ 31.1%

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

   9. Applied egg-rr 48.6%

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

   if 1.5599999999999999e206 < l < 3.2999999999999998e240

   1. Initial program 1.4%

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

   3. Simplified 2.0%

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

   5. Taylor expanded in l around inf 51.8%

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

      1. associate-/l* 51.8%

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

      2. associate-*r/ 51.8%

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

      3. metadata-eval 51.8%

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

   7. Simplified 51.8%

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

   8. Taylor expanded in n around 0 38.3%

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

      1. pow1/2 38.3%

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

      2. associate-*r/ 38.3%

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

      3. *-commutative 38.3%

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

   10. Applied egg-rr 38.3%

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

       1. unpow1/2 38.3%

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

       2. associate-/l* 38.6%

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

       3. *-commutative 38.6%

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

   12. Applied egg-rr 38.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.56 \cdot 10^{+206}:\\ \;\;\;\;{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+240}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{-2}{\frac{Om}{n \cdot U}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{+55} \lor \neg \left(n \leq 5 \cdot 10^{+20}\right):\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -1.75e+55) (not (<= n 5e+20)))
   (pow (* 2.0 (* (* n U) (+ t (* -2.0 (/ (pow l_m 2.0) Om))))) 0.5)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l_m (/ l_m Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.75e+55) || !(n <= 5e+20)) {
		tmp = pow((2.0 * ((n * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om))))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-1.75d+55)) .or. (.not. (n <= 5d+20))) then
        tmp = (2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l_m ** 2.0d0) / om))))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l_m * (l_m / om))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.75e+55) || !(n <= 5e+20)) {
		tmp = Math.pow((2.0 * ((n * U) * (t + (-2.0 * (Math.pow(l_m, 2.0) / Om))))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (n <= -1.75e+55) or not (n <= 5e+20):
		tmp = math.pow((2.0 * ((n * U) * (t + (-2.0 * (math.pow(l_m, 2.0) / Om))))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.75e+55) || !(n <= 5e+20))
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om))))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.75e+55) || ~((n <= 5e+20)))
		tmp = (2.0 * ((n * U) * (t + (-2.0 * ((l_m ^ 2.0) / Om))))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[n, -1.75e+55], N[Not[LessEqual[n, 5e+20]], $MachinePrecision]], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.75000000000000005e55 or 5e20 < n

   1. Initial program 59.1%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in n around 0 40.4%

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

      1. pow1/2 55.0%

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

      2. associate-*r* 58.6%

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

      3. *-commutative 58.6%

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

      4. cancel-sign-sub-inv 58.6%

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

      5. metadata-eval 58.6%

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

   7. Applied egg-rr 58.6%

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

   if -1.75000000000000005e55 < n < 5e20

   1. Initial program 51.0%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in n around 0 52.8%

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

      1. unpow2 52.8%

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

      2. associate-*l/ 59.6%

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

   7. Applied egg-rr 59.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{+55} \lor \neg \left(n \leq 5 \cdot 10^{+20}\right):\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+148}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -4.5e+148)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= t 1.15e+188)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l_m (/ l_m Om))))))))
     (* (sqrt (* 2.0 (* n U))) (sqrt t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -4.5e+148) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (t <= 1.15e+188) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-4.5d+148)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (t <= 1.15d+188) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l_m * (l_m / om))))))))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -4.5e+148) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (t <= 1.15e+188) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -4.5e+148:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif t <= 1.15e+188:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -4.5e+148)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (t <= 1.15e+188)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -4.5e+148)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (t <= 1.15e+188)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -4.5e+148], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t, 1.15e+188], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.49999999999999994e148

   1. Initial program 53.0%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in l around 0 57.9%

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

      1. associate-*r* 58.3%

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

      2. *-commutative 58.3%

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

   7. Simplified 58.3%

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

      1. pow1/2 60.7%

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

      2. associate-*l* 74.3%

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

   9. Applied egg-rr 74.3%

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

   if -4.49999999999999994e148 < t < 1.15000000000000006e188

   1. Initial program 54.5%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in n around 0 48.4%

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

      1. unpow2 48.4%

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

      2. associate-*l/ 53.5%

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

   7. Applied egg-rr 53.5%

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

   if 1.15000000000000006e188 < t

   1. Initial program 51.8%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in l around 0 48.6%

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

      1. associate-*r* 44.4%

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

      2. *-commutative 44.4%

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

   7. Simplified 44.4%

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

      1. pow1/2 52.8%

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

      2. associate-*r* 52.8%

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

      3. associate-*l* 52.8%

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

      4. metadata-eval 52.8%

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

      5. unpow-prod-down 63.8%

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

      6. metadata-eval 63.8%

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

      7. pow1/2 63.8%

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

      8. associate-*l* 63.8%

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

      9. metadata-eval 63.8%

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

      10. pow1/2 63.8%

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

   9. Applied egg-rr 63.8%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+148}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+148}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -2.45e+148)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l_m (/ l_m Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -2.45e+148) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-2.45d+148)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l_m * (l_m / om))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -2.45e+148) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -2.45e+148:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -2.45e+148)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.45e+148)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -2.45e+148], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.45e148

   1. Initial program 53.0%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in l around 0 57.9%

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

      1. associate-*r* 58.3%

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

      2. *-commutative 58.3%

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

   7. Simplified 58.3%

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

      1. pow1/2 60.7%

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

      2. associate-*l* 74.3%

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

   9. Applied egg-rr 74.3%

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

   if -2.45e148 < t

   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 59.7%

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

   5. Taylor expanded in n around 0 47.9%

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

      1. unpow2 47.9%

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

      2. associate-*l/ 52.4%

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

   7. Applied egg-rr 52.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+148}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 9 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 9e-151) (sqrt (* 2.0 (* U (* n t)))) (sqrt (* 2.0 (* t (* n U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 9e-151) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 9d-151) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 9e-151) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 9e-151:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 9e-151)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 9e-151)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 9e-151], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 9.0000000000000005e-151

   1. Initial program 51.1%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in l around 0 41.9%

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

   if 9.0000000000000005e-151 < n

   1. Initial program 59.6%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in l around 0 38.6%

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

      1. associate-*r* 44.1%

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

      2. *-commutative 44.1%

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

   7. Simplified 44.1%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Simplified 57.5%

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

5. Taylor expanded in l around 0 40.7%

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

   1. associate-*r* 38.6%

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

   2. *-commutative 38.6%

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

7. Simplified 38.6%

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

   1. pow1/2 39.7%

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

   2. associate-*l* 43.4%

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

9. Applied egg-rr 43.4%

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

10. Final simplification 43.4%

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

Alternative 14: 35.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Simplified 57.5%

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

5. Taylor expanded in l around 0 40.7%

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

6. Final simplification 40.7%

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

Alternative 15: 34.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Simplified 57.5%

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

5. Taylor expanded in l around 0 40.7%

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

   1. associate-*r* 38.6%

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

   2. *-commutative 38.6%

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

7. Simplified 38.6%

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

   1. pow1/2 39.7%

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

   2. associate-*l* 43.4%

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

9. Applied egg-rr 43.4%

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

    1. unpow1/2 42.2%

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

    2. associate-*r* 42.3%

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

11. Applied egg-rr 42.3%

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

12. Final simplification 42.3%

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

Reproduce

herbie shell --seed 2024030 
(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*))))))