Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 66.6%

Time: 28.9s

Alternatives: 18

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, U \cdot \frac{n \cdot \left(l\_m \cdot l\_m\right)}{Om}, \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot {\left(\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5}\right) \cdot {2}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_1 2e-134)
     (sqrt (fma -4.0 (* U (/ (* n (* l_m l_m)) Om)) (* (* 2.0 U) (* n t))))
     (if (<= t_1 4e+141)
       t_1
       (*
        (*
         l_m
         (pow (* (* n U) (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om))) 0.5))
        (pow 2.0 0.5))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 2e-134) {
		tmp = sqrt(fma(-4.0, (U * ((n * (l_m * l_m)) / Om)), ((2.0 * U) * (n * t))));
	} else if (t_1 <= 4e+141) {
		tmp = t_1;
	} else {
		tmp = (l_m * pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * pow(2.0, 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 2e-134)
		tmp = sqrt(fma(-4.0, Float64(U * Float64(Float64(n * Float64(l_m * l_m)) / Om)), Float64(Float64(2.0 * U) * Float64(n * t))));
	elseif (t_1 <= 4e+141)
		tmp = t_1;
	else
		tmp = Float64(Float64(l_m * (Float64(Float64(n * U) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))) ^ 0.5)) * (2.0 ^ 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-134], N[Sqrt[N[(-4.0 * N[(U * N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 4e+141], t$95$1, N[(N[(l$95$m * N[Power[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000008e-134

   1. Initial program 22.7%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in Om around inf 51.2%

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

      1. fma-define 51.2%

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

      2. associate-/l* 51.2%

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

      3. unpow2 51.2%

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

      4. associate-*r* 51.3%

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

   7. Simplified 51.3%

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

   if 2.00000000000000008e-134 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.00000000000000007e141

   1. Initial program 98.5%

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

   if 4.00000000000000007e141 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 27.6%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in l around inf 27.2%

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

      1. pow1/2 27.3%

         \[\leadsto \color{blue}{{\left(U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}^{0.5}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      2. associate-*r* 29.6%

         \[\leadsto {\color{blue}{\left(\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}^{0.5} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      3. unpow-prod-down 18.6%

         \[\leadsto \color{blue}{\left({\left(U \cdot n\right)}^{0.5} \cdot {\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      4. associate-/l* 18.7%

         \[\leadsto \left({\left(U \cdot n\right)}^{0.5} \cdot {\left(\color{blue}{n \cdot \frac{U* - U}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right) \cdot \left(\ell \cdot \sqrt{2}\right) \]

      5. unpow2 18.7%

         \[\leadsto \left({\left(U \cdot n\right)}^{0.5} \cdot {\left(n \cdot \frac{U* - U}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right) \cdot \left(\ell \cdot \sqrt{2}\right) \]

      6. un-div-inv 18.7%

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

   7. Applied egg-rr 18.7%

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

      1. unpow1/2 17.2%

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

      2. unpow1/2 17.1%

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

      3. unpow2 17.1%

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

      4. associate-*r/ 16.9%

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

      5. unpow2 16.9%

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

      6. times-frac 18.5%

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

   9. Simplified 18.5%

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

       1. associate-*r* 18.5%

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

       2. sqrt-unprod 31.7%

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

       3. pow1/2 32.7%

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

       4. pow1/2 32.7%

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

   11. Applied egg-rr 32.7%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, U \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}, \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {\left(\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5}\right) \cdot {2}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 2.5 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m}{\frac{Om}{l\_m}}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;l\_m \leq 24:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;l\_m \leq 3.9 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 3.3 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t - \left(\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) - -2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot {\left(\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5}\right) \cdot {2}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 2.5e-198)
     (sqrt
      (*
       t_1
       (+
        (- t (* 2.0 (/ l_m (/ Om l_m))))
        (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
     (if (<= l_m 24.0)
       (sqrt
        (*
         (* 2.0 U)
         (*
          n
          (-
           t
           (-
            (/ (* 2.0 (* l_m l_m)) Om)
            (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
       (if (<= l_m 3.9e+67)
         (sqrt
          (*
           t_1
           (+ t (* (* l_m l_m) (- (* n (/ (- U* U) (* Om Om))) (/ 2.0 Om))))))
         (if (<= l_m 3.3e+157)
           (sqrt
            (*
             U
             (*
              (* 2.0 n)
              (-
               t
               (-
                (* (* n (- U U*)) (* (/ l_m Om) (/ l_m Om)))
                (* -2.0 (* l_m (/ l_m Om))))))))
           (*
            (*
             l_m
             (pow (* (* n U) (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om))) 0.5))
            (pow 2.0 0.5))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 2.5e-198) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l_m / (Om / l_m)))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	} else if (l_m <= 24.0) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else if (l_m <= 3.9e+67) {
		tmp = sqrt((t_1 * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))));
	} else if (l_m <= 3.3e+157) {
		tmp = sqrt((U * ((2.0 * n) * (t - (((n * (U - U_42_)) * ((l_m / Om) * (l_m / Om))) - (-2.0 * (l_m * (l_m / Om))))))));
	} else {
		tmp = (l_m * pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * pow(2.0, 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (l_m <= 2.5d-198) then
        tmp = sqrt((t_1 * ((t - (2.0d0 * (l_m / (om / l_m)))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    else if (l_m <= 24.0d0) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else if (l_m <= 3.9d+67) then
        tmp = sqrt((t_1 * (t + ((l_m * l_m) * ((n * ((u_42 - u) / (om * om))) - (2.0d0 / om))))))
    else if (l_m <= 3.3d+157) then
        tmp = sqrt((u * ((2.0d0 * n) * (t - (((n * (u - u_42)) * ((l_m / om) * (l_m / om))) - ((-2.0d0) * (l_m * (l_m / om))))))))
    else
        tmp = (l_m * (((n * u) * (((n / om) * ((u_42 - u) / om)) - (2.0d0 / om))) ** 0.5d0)) * (2.0d0 ** 0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 2.5e-198) {
		tmp = Math.sqrt((t_1 * ((t - (2.0 * (l_m / (Om / l_m)))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	} else if (l_m <= 24.0) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else if (l_m <= 3.9e+67) {
		tmp = Math.sqrt((t_1 * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))));
	} else if (l_m <= 3.3e+157) {
		tmp = Math.sqrt((U * ((2.0 * n) * (t - (((n * (U - U_42_)) * ((l_m / Om) * (l_m / Om))) - (-2.0 * (l_m * (l_m / Om))))))));
	} else {
		tmp = (l_m * Math.pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * Math.pow(2.0, 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if l_m <= 2.5e-198:
		tmp = math.sqrt((t_1 * ((t - (2.0 * (l_m / (Om / l_m)))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	elif l_m <= 24.0:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	elif l_m <= 3.9e+67:
		tmp = math.sqrt((t_1 * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))))
	elif l_m <= 3.3e+157:
		tmp = math.sqrt((U * ((2.0 * n) * (t - (((n * (U - U_42_)) * ((l_m / Om) * (l_m / Om))) - (-2.0 * (l_m * (l_m / Om))))))))
	else:
		tmp = (l_m * math.pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * math.pow(2.0, 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 2.5e-198)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l_m / Float64(Om / l_m)))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))));
	elseif (l_m <= 24.0)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	elseif (l_m <= 3.9e+67)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(n * Float64(Float64(U_42_ - U) / Float64(Om * Om))) - Float64(2.0 / Om))))));
	elseif (l_m <= 3.3e+157)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t - Float64(Float64(Float64(n * Float64(U - U_42_)) * Float64(Float64(l_m / Om) * Float64(l_m / Om))) - Float64(-2.0 * Float64(l_m * Float64(l_m / Om))))))));
	else
		tmp = Float64(Float64(l_m * (Float64(Float64(n * U) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))) ^ 0.5)) * (2.0 ^ 0.5));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (l_m <= 2.5e-198)
		tmp = sqrt((t_1 * ((t - (2.0 * (l_m / (Om / l_m)))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	elseif (l_m <= 24.0)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	elseif (l_m <= 3.9e+67)
		tmp = sqrt((t_1 * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))));
	elseif (l_m <= 3.3e+157)
		tmp = sqrt((U * ((2.0 * n) * (t - (((n * (U - U_42_)) * ((l_m / Om) * (l_m / Om))) - (-2.0 * (l_m * (l_m / Om))))))));
	else
		tmp = (l_m * (((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))) ^ 0.5)) * (2.0 ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 2.5e-198], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l$95$m / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 24.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.9e+67], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.3e+157], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t - N[(N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Power[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 2.5e-198

   1. Initial program 55.9%

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

      1. associate-/l* 57.6%

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

      2. clear-num 57.6%

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

      3. un-div-inv 57.6%

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

   4. Applied egg-rr 57.6%

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

   if 2.5e-198 < l < 24

   1. Initial program 62.0%

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

   3. Taylor expanded in U around 0 63.8%

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

      1. associate-*r* 63.9%

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

      2. +-commutative 63.9%

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

      3. mul-1-neg 63.9%

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

      4. unsub-neg 63.9%

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

      5. associate-*r/ 63.9%

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

      6. unpow2 63.9%

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

      7. associate-/l* 63.9%

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

      8. unpow2 63.9%

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

      9. times-frac 66.4%

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

      10. unpow2 66.4%

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

   5. Simplified 66.4%

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

   if 24 < l < 3.90000000000000007e67

   1. Initial program 61.7%

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

      1. associate-/l* 61.7%

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

      2. clear-num 61.7%

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

      3. un-div-inv 61.7%

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

   4. Applied egg-rr 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

   6. Applied egg-rr 61.7%

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

      1. associate-*r* 61.0%

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

   8. Simplified 61.0%

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

   9. Taylor expanded in l around 0 71.0%

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

       1. mul-1-neg 71.0%

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

       2. unpow2 71.0%

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

       3. associate-*r/ 71.0%

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

       4. metadata-eval 71.0%

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

       5. associate-/l* 71.7%

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

       6. unpow2 71.7%

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

   11. Simplified 71.7%

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

   if 3.90000000000000007e67 < l < 3.3000000000000002e157

   1. Initial program 66.2%

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

      1. associate-*r* 66.2%

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

      2. *-commutative 66.2%

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

      3. associate-*r* 66.2%

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

      4. associate-*r* 77.5%

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

   4. Applied egg-rr 77.5%

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

   if 3.3000000000000002e157 < l

   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 42.3%

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

   5. Taylor expanded in l around inf 57.0%

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

      1. pow1/2 57.2%

         \[\leadsto \color{blue}{{\left(U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}^{0.5}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      2. associate-*r* 65.9%

         \[\leadsto {\color{blue}{\left(\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}^{0.5} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      3. unpow-prod-down 43.7%

         \[\leadsto \color{blue}{\left({\left(U \cdot n\right)}^{0.5} \cdot {\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      4. associate-/l* 41.1%

         \[\leadsto \left({\left(U \cdot n\right)}^{0.5} \cdot {\left(\color{blue}{n \cdot \frac{U* - U}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right) \cdot \left(\ell \cdot \sqrt{2}\right) \]

      5. unpow2 41.1%

         \[\leadsto \left({\left(U \cdot n\right)}^{0.5} \cdot {\left(n \cdot \frac{U* - U}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right) \cdot \left(\ell \cdot \sqrt{2}\right) \]

      6. un-div-inv 41.1%

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

   7. Applied egg-rr 41.1%

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

      1. unpow1/2 41.1%

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

      2. unpow1/2 41.0%

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

      3. unpow2 41.0%

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

      4. associate-*r/ 43.6%

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

      5. unpow2 43.6%

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

      6. times-frac 49.4%

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

   9. Simplified 49.4%

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

       1. associate-*r* 49.5%

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

       2. sqrt-unprod 74.0%

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

       3. pow1/2 74.0%

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

       4. pow1/2 74.0%

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

   11. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{\left({\left(\left(U \cdot n\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5} \cdot \ell\right) \cdot {2}^{0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 24:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - U* \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t - \left(\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) - -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {\left(\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5}\right) \cdot {2}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.75 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot {\left(\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5}\right) \cdot {2}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.75e-206)
   (sqrt (fabs (* 2.0 (* t (* n U)))))
   (if (<= l_m 9.5e+158)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (-
         t
         (-
          (/ (* 2.0 (* l_m l_m)) Om)
          (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
     (*
      (* l_m (pow (* (* n U) (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om))) 0.5))
      (pow 2.0 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.75e-206) {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	} else if (l_m <= 9.5e+158) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = (l_m * pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * pow(2.0, 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.75d-206) then
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    else if (l_m <= 9.5d+158) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else
        tmp = (l_m * (((n * u) * (((n / om) * ((u_42 - u) / om)) - (2.0d0 / om))) ** 0.5d0)) * (2.0d0 ** 0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.75e-206) {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	} else if (l_m <= 9.5e+158) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = (l_m * Math.pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * Math.pow(2.0, 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.75e-206:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	elif l_m <= 9.5e+158:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	else:
		tmp = (l_m * math.pow(((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))), 0.5)) * math.pow(2.0, 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.75e-206)
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	elseif (l_m <= 9.5e+158)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	else
		tmp = Float64(Float64(l_m * (Float64(Float64(n * U) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))) ^ 0.5)) * (2.0 ^ 0.5));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.75e-206)
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	elseif (l_m <= 9.5e+158)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	else
		tmp = (l_m * (((n * U) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))) ^ 0.5)) * (2.0 ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.75e-206], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 9.5e+158], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Power[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.74999999999999995e-206

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in l around 0 37.4%

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

      1. associate-*r* 37.4%

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

   7. Simplified 37.4%

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

      1. add-sqr-sqrt 37.4%

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

      2. pow1/2 37.4%

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

      3. pow1/2 39.5%

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

      4. pow-prod-down 22.1%

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

      5. associate-*l* 22.1%

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

      6. associate-*l* 22.1%

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

   9. Applied egg-rr 22.1%

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

       1. unpow1/2 22.1%

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

       2. rem-sqrt-square 40.0%

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

       3. associate-*r* 43.4%

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

   11. Simplified 43.4%

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

   if 1.74999999999999995e-206 < l < 9.49999999999999913e158

   1. Initial program 61.8%

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

   3. Taylor expanded in U around 0 61.3%

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

      1. associate-*r* 61.4%

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

      2. +-commutative 61.4%

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

      3. mul-1-neg 61.4%

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

      4. unsub-neg 61.4%

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

      5. associate-*r/ 61.4%

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

      6. unpow2 61.4%

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

      7. associate-/l* 64.2%

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

      8. unpow2 64.2%

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

      9. times-frac 65.9%

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

      10. unpow2 65.9%

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

   5. Simplified 65.9%

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

   if 9.49999999999999913e158 < l

   1. Initial program 15.0%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in l around inf 57.5%

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

      1. pow1/2 57.6%

         \[\leadsto \color{blue}{{\left(U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}^{0.5}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      2. associate-*r* 66.9%

         \[\leadsto {\color{blue}{\left(\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}^{0.5} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      3. unpow-prod-down 43.3%

         \[\leadsto \color{blue}{\left({\left(U \cdot n\right)}^{0.5} \cdot {\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      4. associate-/l* 40.6%

         \[\leadsto \left({\left(U \cdot n\right)}^{0.5} \cdot {\left(\color{blue}{n \cdot \frac{U* - U}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right) \cdot \left(\ell \cdot \sqrt{2}\right) \]

      5. unpow2 40.6%

         \[\leadsto \left({\left(U \cdot n\right)}^{0.5} \cdot {\left(n \cdot \frac{U* - U}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)}^{0.5}\right) \cdot \left(\ell \cdot \sqrt{2}\right) \]

      6. un-div-inv 40.6%

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

   7. Applied egg-rr 40.6%

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

      1. unpow1/2 40.6%

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

      2. unpow1/2 40.4%

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

      3. unpow2 40.4%

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

      4. associate-*r/ 43.2%

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

      5. unpow2 43.2%

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

      6. times-frac 49.3%

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

   9. Simplified 49.3%

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

       1. associate-*r* 49.4%

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

       2. sqrt-unprod 75.4%

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

       3. pow1/2 75.4%

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

       4. pow1/2 75.4%

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

   11. Applied egg-rr 75.4%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - U* \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {\left(\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}^{0.5}\right) \cdot {2}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9.4 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 1.45 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 9.4e-207)
   (sqrt (fabs (* 2.0 (* t (* n U)))))
   (if (<= l_m 1.45e+152)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (-
         t
         (-
          (/ (* 2.0 (* l_m l_m)) Om)
          (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
     (*
      (sqrt (* U (* n (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om)))))
      (* l_m (sqrt 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9.4e-207) {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	} else if (l_m <= 1.45e+152) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 9.4d-207) then
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    else if (l_m <= 1.45d+152) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else
        tmp = sqrt((u * (n * (((n / om) * ((u_42 - u) / om)) - (2.0d0 / om))))) * (l_m * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9.4e-207) {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	} else if (l_m <= 1.45e+152) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = Math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) - (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_):
	tmp = 0
	if l_m <= 9.4e-207:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	elif l_m <= 1.45e+152:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	else:
		tmp = math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) - (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_)
	tmp = 0.0
	if (l_m <= 9.4e-207)
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	elseif (l_m <= 1.45e+152)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))))) * Float64(l_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_)
	tmp = 0.0;
	if (l_m <= 9.4e-207)
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	elseif (l_m <= 1.45e+152)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	else
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) - (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$42$_] := If[LessEqual[l$95$m, 9.4e-207], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.45e+152], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.40000000000000057e-207

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in l around 0 37.4%

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

      1. associate-*r* 37.4%

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

   7. Simplified 37.4%

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

      1. add-sqr-sqrt 37.4%

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

      2. pow1/2 37.4%

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

      3. pow1/2 39.5%

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

      4. pow-prod-down 22.1%

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

      5. associate-*l* 22.1%

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

      6. associate-*l* 22.1%

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

   9. Applied egg-rr 22.1%

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

       1. unpow1/2 22.1%

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

       2. rem-sqrt-square 40.0%

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

       3. associate-*r* 43.4%

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

   11. Simplified 43.4%

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

   if 9.40000000000000057e-207 < l < 1.4499999999999999e152

   1. Initial program 62.1%

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

   3. Taylor expanded in U around 0 61.6%

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

      1. associate-*r* 61.6%

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

      2. +-commutative 61.6%

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

      3. mul-1-neg 61.6%

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

      4. unsub-neg 61.6%

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

      5. associate-*r/ 61.6%

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

      6. unpow2 61.6%

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

      7. associate-/l* 64.5%

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

      8. unpow2 64.5%

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

      9. times-frac 66.3%

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

      10. unpow2 66.3%

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

   5. Simplified 66.3%

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

   if 1.4499999999999999e152 < l

   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 42.3%

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

   5. Taylor expanded in l around inf 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-/l* 57.0%

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

      3. unpow2 57.0%

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

      4. un-div-inv 57.0%

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

   7. Applied egg-rr 57.0%

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

      1. unpow2 57.0%

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

      2. div-sub 39.3%

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

      3. metadata-eval 39.3%

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

      4. associate-*r/ 39.3%

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

      5. div-sub 57.0%

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

      6. associate-*r/ 57.0%

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

      7. unpow2 57.0%

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

      8. times-frac 68.1%

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

      9. associate-*r/ 68.1%

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

      10. metadata-eval 68.1%

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

   9. Simplified 68.1%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.4 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - U* \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.3e-206)
   (sqrt (fabs (* 2.0 (* t (* n U)))))
   (if (<= l_m 9.5e+158)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (-
         t
         (-
          (/ (* 2.0 (* l_m l_m)) Om)
          (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
     (*
      (* l_m (sqrt 2.0))
      (sqrt (* (* n U) (- (* n (/ (- U* U) (* Om Om))) (/ 2.0 Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.3e-206) {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	} else if (l_m <= 9.5e+158) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.3d-206) then
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    else if (l_m <= 9.5d+158) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * u) * ((n * ((u_42 - u) / (om * om))) - (2.0d0 / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.3e-206) {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	} else if (l_m <= 9.5e+158) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * U) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.3e-206:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	elif l_m <= 9.5e+158:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * U) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.3e-206)
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	elseif (l_m <= 9.5e+158)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(n * Float64(Float64(U_42_ - U) / Float64(Om * Om))) - 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_)
	tmp = 0.0;
	if (l_m <= 1.3e-206)
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	elseif (l_m <= 9.5e+158)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.3e-206], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 9.5e+158], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.3e-206

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in l around 0 37.4%

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

      1. associate-*r* 37.4%

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

   7. Simplified 37.4%

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

      1. add-sqr-sqrt 37.4%

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

      2. pow1/2 37.4%

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

      3. pow1/2 39.5%

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

      4. pow-prod-down 22.1%

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

      5. associate-*l* 22.1%

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

      6. associate-*l* 22.1%

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

   9. Applied egg-rr 22.1%

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

       1. unpow1/2 22.1%

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

       2. rem-sqrt-square 40.0%

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

       3. associate-*r* 43.4%

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

   11. Simplified 43.4%

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

   if 1.3e-206 < l < 9.49999999999999913e158

   1. Initial program 61.8%

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

   3. Taylor expanded in U around 0 61.3%

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

      1. associate-*r* 61.4%

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

      2. +-commutative 61.4%

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

      3. mul-1-neg 61.4%

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

      4. unsub-neg 61.4%

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

      5. associate-*r/ 61.4%

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

      6. unpow2 61.4%

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

      7. associate-/l* 64.2%

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

      8. unpow2 64.2%

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

      9. times-frac 65.9%

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

      10. unpow2 65.9%

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

   5. Simplified 65.9%

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

   if 9.49999999999999913e158 < l

   1. Initial program 15.0%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in l around inf 57.5%

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

      1. associate-*r* 66.8%

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

      2. associate-/l* 66.7%

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

      3. unpow2 66.7%

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

      4. un-div-inv 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - U* \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;l\_m \leq 2.2 \cdot 10^{+275}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3e-198)
   (sqrt (fabs (* 2.0 (* t (* n U)))))
   (if (<= l_m 2.6e+162)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (-
         t
         (-
          (/ (* 2.0 (* l_m l_m)) Om)
          (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
     (if (<= l_m 2.2e+275)
       (pow (* (* 2.0 U) (+ (* n t) (* n (* -2.0 (* l_m (/ l_m Om)))))) 0.5)
       (* (* l_m (sqrt 2.0)) (sqrt (* -2.0 (/ (* n U) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3e-198) {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	} else if (l_m <= 2.6e+162) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else if (l_m <= 2.2e+275) {
		tmp = pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * ((n * U) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 3d-198) then
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    else if (l_m <= 2.6d+162) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else if (l_m <= 2.2d+275) then
        tmp = ((2.0d0 * u) * ((n * t) + (n * ((-2.0d0) * (l_m * (l_m / om)))))) ** 0.5d0
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((-2.0d0) * ((n * u) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3e-198) {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	} else if (l_m <= 2.6e+162) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else if (l_m <= 2.2e+275) {
		tmp = Math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((-2.0 * ((n * U) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 3e-198:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	elif l_m <= 2.6e+162:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	elif l_m <= 2.2e+275:
		tmp = math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5)
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((-2.0 * ((n * U) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3e-198)
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	elseif (l_m <= 2.6e+162)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	elseif (l_m <= 2.2e+275)
		tmp = Float64(Float64(2.0 * U) * Float64(Float64(n * t) + Float64(n * Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 0.5;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(-2.0 * Float64(Float64(n * U) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 3e-198)
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	elseif (l_m <= 2.6e+162)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	elseif (l_m <= 2.2e+275)
		tmp = ((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))) ^ 0.5;
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((-2.0 * ((n * U) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3e-198], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.6e+162], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.2e+275], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(n * t), $MachinePrecision] + N[(n * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.0000000000000001e-198

   1. Initial program 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in l around 0 37.2%

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

      1. associate-*r* 37.2%

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

   7. Simplified 37.2%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow1/2 37.2%

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

      3. pow1/2 39.2%

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

      4. pow-prod-down 22.0%

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

      5. associate-*l* 22.0%

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

      6. associate-*l* 22.0%

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

   9. Applied egg-rr 22.0%

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

       1. unpow1/2 22.0%

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

       2. rem-sqrt-square 39.7%

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

       3. associate-*r* 43.1%

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

   11. Simplified 43.1%

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

   if 3.0000000000000001e-198 < l < 2.6e162

   1. Initial program 62.3%

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

   3. Taylor expanded in U around 0 61.9%

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

      1. associate-*r* 61.9%

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

      2. +-commutative 61.9%

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

      3. mul-1-neg 61.9%

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

      4. unsub-neg 61.9%

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

      5. associate-*r/ 61.9%

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

      6. unpow2 61.9%

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

      7. associate-/l* 64.7%

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

      8. unpow2 64.7%

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

      9. times-frac 66.2%

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

      10. unpow2 66.2%

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

   5. Simplified 66.2%

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

   if 2.6e162 < l < 2.1999999999999999e275

   1. Initial program 11.5%

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

      1. associate-/l* 49.3%

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

      2. clear-num 49.4%

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

      3. un-div-inv 49.4%

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

   4. Applied egg-rr 49.4%

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

   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. cancel-sign-sub-inv 12.1%

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

      2. metadata-eval 12.1%

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

      3. unpow2 12.1%

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

      4. associate-*r/ 45.6%

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

   7. Simplified 45.6%

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

      1. pow1/2 59.4%

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

      2. associate-*r* 59.4%

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

      3. distribute-lft-in 55.1%

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

   9. Applied egg-rr 55.1%

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

   if 2.1999999999999999e275 < l

   1. Initial program 16.5%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in l around inf 43.8%

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

   6. Taylor expanded in n around 0 57.6%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - U* \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+275}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{-2 \cdot \frac{n \cdot U}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.8e-198)
   (sqrt (fabs (* 2.0 (* t (* n U)))))
   (if (<= l_m 2.6e+162)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (-
         t
         (-
          (/ (* 2.0 (* l_m l_m)) Om)
          (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
     (pow (* (* 2.0 U) (+ (* n t) (* n (* -2.0 (* l_m (/ l_m Om)))))) 0.5))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.8e-198) {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	} else if (l_m <= 2.6e+162) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.8d-198) then
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    else if (l_m <= 2.6d+162) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else
        tmp = ((2.0d0 * u) * ((n * t) + (n * ((-2.0d0) * (l_m * (l_m / om)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.8e-198) {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	} else if (l_m <= 2.6e+162) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = Math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.8e-198:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	elif l_m <= 2.6e+162:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	else:
		tmp = math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.8e-198)
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	elseif (l_m <= 2.6e+162)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	else
		tmp = Float64(Float64(2.0 * U) * Float64(Float64(n * t) + Float64(n * Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.8e-198)
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	elseif (l_m <= 2.6e+162)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	else
		tmp = ((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.8e-198], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.6e+162], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(n * t), $MachinePrecision] + N[(n * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.79999999999999999e-198

   1. Initial program 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in l around 0 37.2%

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

      1. associate-*r* 37.2%

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

   7. Simplified 37.2%

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

      1. add-sqr-sqrt 37.2%

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

      2. pow1/2 37.2%

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

      3. pow1/2 39.2%

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

      4. pow-prod-down 22.0%

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

      5. associate-*l* 22.0%

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

      6. associate-*l* 22.0%

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

   9. Applied egg-rr 22.0%

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

       1. unpow1/2 22.0%

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

       2. rem-sqrt-square 39.7%

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

       3. associate-*r* 43.1%

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

   11. Simplified 43.1%

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

   if 1.79999999999999999e-198 < l < 2.6e162

   1. Initial program 62.3%

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

   3. Taylor expanded in U around 0 61.9%

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

      1. associate-*r* 61.9%

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

      2. +-commutative 61.9%

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

      3. mul-1-neg 61.9%

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

      4. unsub-neg 61.9%

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

      5. associate-*r/ 61.9%

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

      6. unpow2 61.9%

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

      7. associate-/l* 64.7%

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

      8. unpow2 64.7%

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

      9. times-frac 66.2%

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

      10. unpow2 66.2%

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

   5. Simplified 66.2%

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

   if 2.6e162 < l

   1. Initial program 12.7%

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

      1. associate-/l* 41.7%

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

      2. clear-num 41.7%

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

      3. un-div-inv 41.7%

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

   4. Applied egg-rr 41.7%

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

   5. Taylor expanded in n around 0 13.6%

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

      1. cancel-sign-sub-inv 13.6%

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

      2. metadata-eval 13.6%

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

      3. unpow2 13.6%

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

      4. associate-*r/ 39.2%

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

   7. Simplified 39.2%

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

      1. pow1/2 49.8%

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

      2. associate-*r* 49.8%

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

      3. distribute-lft-in 43.2%

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

   9. Applied egg-rr 43.2%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om} - U* \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.7 \cdot 10^{-180}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;l\_m \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om} - U* \cdot \left(\frac{l\_m \cdot l\_m}{Om} \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.7e-180)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (if (<= l_m 2.6e+162)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (-
         t
         (-
          (/ (* 2.0 (* l_m l_m)) Om)
          (* U* (* (/ (* l_m l_m) Om) (/ n Om))))))))
     (pow (* (* 2.0 U) (+ (* n t) (* n (* -2.0 (* l_m (/ l_m Om)))))) 0.5))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.7e-180) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else if (l_m <= 2.6e+162) {
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.7d-180) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else if (l_m <= 2.6d+162) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (((2.0d0 * (l_m * l_m)) / om) - (u_42 * (((l_m * l_m) / om) * (n / om))))))))
    else
        tmp = ((2.0d0 * u) * ((n * t) + (n * ((-2.0d0) * (l_m * (l_m / om)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.7e-180) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else if (l_m <= 2.6e+162) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	} else {
		tmp = Math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.7e-180:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	elif l_m <= 2.6e+162:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))))
	else:
		tmp = math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.7e-180)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	elseif (l_m <= 2.6e+162)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) / Om) - Float64(U_42_ * Float64(Float64(Float64(l_m * l_m) / Om) * Float64(n / Om))))))));
	else
		tmp = Float64(Float64(2.0 * U) * Float64(Float64(n * t) + Float64(n * Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.7e-180)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	elseif (l_m <= 2.6e+162)
		tmp = sqrt(((2.0 * U) * (n * (t - (((2.0 * (l_m * l_m)) / Om) - (U_42_ * (((l_m * l_m) / Om) * (n / Om))))))));
	else
		tmp = ((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.7e-180], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 2.6e+162], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(n * t), $MachinePrecision] + N[(n * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.69999999999999991e-180

   1. Initial program 56.2%

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

      1. associate-/l* 57.9%

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

      2. clear-num 57.9%

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

      3. un-div-inv 57.9%

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

   4. Applied egg-rr 57.9%

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

      1. *-commutative 57.9%

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

      2. unpow2 57.9%

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

   6. Applied egg-rr 57.9%

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

      1. associate-*r* 54.4%

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

   8. Simplified 54.4%

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

   9. Taylor expanded in t around inf 40.9%

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

       1. pow1/2 42.9%

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

       2. *-commutative 42.9%

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

       3. associate-*l* 42.9%

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

   11. Applied egg-rr 42.9%

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

   if 1.69999999999999991e-180 < l < 2.6e162

   1. Initial program 61.7%

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

   3. Taylor expanded in U around 0 61.3%

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

      1. associate-*r* 61.4%

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

      2. +-commutative 61.4%

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

      3. mul-1-neg 61.4%

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

      4. unsub-neg 61.4%

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

      5. associate-*r/ 61.4%

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

      6. unpow2 61.4%

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

      7. associate-/l* 64.2%

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

      8. unpow2 64.2%

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

      9. times-frac 65.8%

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

      10. unpow2 65.8%

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

   5. Simplified 65.8%

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

   if 2.6e162 < l

   1. Initial program 12.7%

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

      1. associate-/l* 41.7%

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

      2. clear-num 41.7%

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

      3. un-div-inv 41.7%

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

   4. Applied egg-rr 41.7%

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

   5. Taylor expanded in n around 0 13.6%

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

      1. cancel-sign-sub-inv 13.6%

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

      2. metadata-eval 13.6%

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

      3. unpow2 13.6%

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

      4. associate-*r/ 39.2%

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

   7. Simplified 39.2%

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

      1. pow1/2 49.8%

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

      2. associate-*r* 49.8%

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

      3. distribute-lft-in 43.2%

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

   9. Applied egg-rr 43.2%

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

Alternative 9: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+37} \lor \neg \left(n \leq 7.5 \cdot 10^{-90}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -3.7e+37) (not (<= n 7.5e-90)))
   (sqrt
    (*
     (* (* 2.0 n) U)
     (+ t (* (* l_m l_m) (- (* n (/ (- U* U) (* Om Om))) (/ 2.0 Om))))))
   (pow (* (* 2.0 U) (+ (* n t) (* n (* -2.0 (* l_m (/ l_m Om)))))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -3.7e+37) || !(n <= 7.5e-90)) {
		tmp = sqrt((((2.0 * n) * U) * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))));
	} else {
		tmp = pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-3.7d+37)) .or. (.not. (n <= 7.5d-90))) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((l_m * l_m) * ((n * ((u_42 - u) / (om * om))) - (2.0d0 / om))))))
    else
        tmp = ((2.0d0 * u) * ((n * t) + (n * ((-2.0d0) * (l_m * (l_m / om)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -3.7e+37) || !(n <= 7.5e-90)) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))));
	} else {
		tmp = Math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (n <= -3.7e+37) or not (n <= 7.5e-90):
		tmp = math.sqrt((((2.0 * n) * U) * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))))
	else:
		tmp = math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((n <= -3.7e+37) || !(n <= 7.5e-90))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(l_m * l_m) * Float64(Float64(n * Float64(Float64(U_42_ - U) / Float64(Om * Om))) - Float64(2.0 / Om))))));
	else
		tmp = Float64(Float64(2.0 * U) * Float64(Float64(n * t) + Float64(n * Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((n <= -3.7e+37) || ~((n <= 7.5e-90)))
		tmp = sqrt((((2.0 * n) * U) * (t + ((l_m * l_m) * ((n * ((U_42_ - U) / (Om * Om))) - (2.0 / Om))))));
	else
		tmp = ((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[n, -3.7e+37], N[Not[LessEqual[n, 7.5e-90]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(n * t), $MachinePrecision] + N[(n * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.6999999999999999e37 or 7.4999999999999999e-90 < n

   1. Initial program 60.8%

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

      1. associate-/l* 64.0%

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

      2. clear-num 64.0%

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

      3. un-div-inv 64.0%

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

   4. Applied egg-rr 64.0%

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

      1. *-commutative 64.0%

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

      2. unpow2 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. associate-*r* 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. unpow2 57.1%

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

       3. associate-*r/ 57.1%

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

       4. metadata-eval 57.1%

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

       5. associate-/l* 63.4%

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

       6. unpow2 63.4%

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

   11. Simplified 63.4%

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

   if -3.6999999999999999e37 < n < 7.4999999999999999e-90

   1. Initial program 45.5%

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

      1. associate-/l* 51.0%

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

      2. clear-num 51.0%

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

      3. un-div-inv 51.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in n around 0 49.0%

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

      1. cancel-sign-sub-inv 49.0%

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

      2. metadata-eval 49.0%

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

      3. unpow2 49.0%

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

      4. associate-*r/ 55.5%

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

   7. Simplified 55.5%

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

      1. pow1/2 56.4%

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

      2. associate-*r* 56.4%

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

      3. distribute-lft-in 56.4%

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

   9. Applied egg-rr 56.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+37} \lor \neg \left(n \leq 7.5 \cdot 10^{-90}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{U* - U}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t + n \cdot \left(-2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -8e+46)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (pow (* (* 2.0 U) (+ (* n t) (* n (* -2.0 (* l_m (/ l_m Om)))))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -8e+46) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-8d+46)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = ((2.0d0 * u) * ((n * t) + (n * ((-2.0d0) * (l_m * (l_m / om)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -8e+46) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -8e+46:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.pow(((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -8e+46)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * U) * Float64(Float64(n * t) + Float64(n * Float64(-2.0 * Float64(l_m * Float64(l_m / Om)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -8e+46)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = ((2.0 * U) * ((n * t) + (n * (-2.0 * (l_m * (l_m / Om)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -8e+46], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(n * t), $MachinePrecision] + N[(n * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.9999999999999999e46

   1. Initial program 53.4%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in l around 0 40.3%

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

      1. associate-*r* 40.3%

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

   7. Simplified 40.3%

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

      1. pow1/2 51.3%

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

      2. associate-*l* 51.3%

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

   9. Applied egg-rr 51.3%

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

   if -7.9999999999999999e46 < t

   1. Initial program 52.4%

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

      1. associate-/l* 57.1%

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

      2. clear-num 57.1%

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

      3. un-div-inv 57.1%

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

   4. Applied egg-rr 57.1%

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

   5. Taylor expanded in n around 0 48.0%

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

      1. cancel-sign-sub-inv 48.0%

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

      2. metadata-eval 48.0%

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

      3. unpow2 48.0%

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

      4. associate-*r/ 52.9%

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

   7. Simplified 52.9%

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

      1. pow1/2 57.0%

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

      2. associate-*r* 57.0%

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

      3. distribute-lft-in 56.0%

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

   9. Applied egg-rr 56.0%

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

Alternative 11: 49.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9.4 \cdot 10^{-207}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 9.4e-207)
   (pow (* t (* 2.0 (* n U))) 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 (l_m <= 9.4e-207) {
		tmp = pow((t * (2.0 * (n * U))), 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 (l_m <= 9.4d-207) then
        tmp = (t * (2.0d0 * (n * u))) ** 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 (l_m <= 9.4e-207) {
		tmp = Math.pow((t * (2.0 * (n * U))), 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 l_m <= 9.4e-207:
		tmp = math.pow((t * (2.0 * (n * U))), 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 (l_m <= 9.4e-207)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 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 (l_m <= 9.4e-207)
		tmp = (t * (2.0 * (n * U))) ^ 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[l$95$m, 9.4e-207], N[Power[N[(t * N[(2.0 * N[(n * U), $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 l < 9.40000000000000057e-207

   1. Initial program 56.2%

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

      1. associate-/l* 57.9%

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

      2. clear-num 57.9%

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

      3. un-div-inv 57.9%

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

   4. Applied egg-rr 57.9%

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

      1. *-commutative 57.9%

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

      2. unpow2 57.9%

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

   6. Applied egg-rr 57.9%

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

      1. associate-*r* 55.0%

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

   8. Simplified 55.0%

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

   9. Taylor expanded in t around inf 40.8%

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

       1. pow1/2 42.8%

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

       2. *-commutative 42.8%

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

       3. associate-*l* 42.8%

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

   11. Applied egg-rr 42.8%

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

   if 9.40000000000000057e-207 < l

   1. Initial program 47.5%

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

      1. associate-/l* 55.8%

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

      2. clear-num 55.8%

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

      3. un-div-inv 55.8%

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

   4. Applied egg-rr 55.8%

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

   5. Taylor expanded in n around 0 43.5%

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

      1. cancel-sign-sub-inv 43.5%

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

      2. metadata-eval 43.5%

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

      3. unpow2 43.5%

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

      4. associate-*r/ 50.8%

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

   7. Simplified 50.8%

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

Alternative 12: 43.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9 \cdot 10^{+83}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 9e+83)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (sqrt (* 2.0 (* U (* -2.0 (* l_m (* l_m (/ n Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9e+83) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (-2.0 * (l_m * (l_m * (n / Om)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 9d+83) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * ((-2.0d0) * (l_m * (l_m * (n / om)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9e+83) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (-2.0 * (l_m * (l_m * (n / Om)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 9e+83:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (-2.0 * (l_m * (l_m * (n / Om)))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 9e+83)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(-2.0 * Float64(l_m * Float64(l_m * Float64(n / Om)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 9e+83)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (-2.0 * (l_m * (l_m * (n / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 9e+83], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(-2.0 * N[(l$95$m * N[(l$95$m * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.9999999999999999e83

   1. Initial program 57.6%

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

      1. associate-/l* 58.8%

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

      2. clear-num 58.8%

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

      3. un-div-inv 58.8%

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

   4. Applied egg-rr 58.8%

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

      1. *-commutative 58.8%

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

      2. unpow2 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. associate-*r* 55.2%

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

   8. Simplified 55.2%

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

   9. Taylor expanded in t around inf 43.4%

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

       1. pow1/2 45.4%

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

       2. *-commutative 45.4%

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

       3. associate-*l* 45.4%

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

   11. Applied egg-rr 45.4%

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

   if 8.9999999999999999e83 < l

   1. Initial program 31.3%

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

      1. associate-/l* 49.5%

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

      2. clear-num 49.5%

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

      3. un-div-inv 49.5%

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

   4. Applied egg-rr 49.5%

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

   5. Taylor expanded in n around 0 31.6%

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

      1. cancel-sign-sub-inv 31.6%

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

      2. metadata-eval 31.6%

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

      3. unpow2 31.6%

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

      4. associate-*r/ 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in t around 0 31.5%

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

      1. associate-*r/ 23.6%

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

      2. unpow2 23.6%

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

      3. associate-*r* 36.0%

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

   10. Simplified 36.0%

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

Alternative 13: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 5.5e+204)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 5.5e+204) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 5.5d+204) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 5.5e+204) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 5.5e+204:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 5.5e+204)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= 5.5e+204)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 5.5e+204], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.4999999999999996e204

   1. Initial program 53.5%

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

      1. associate-/l* 57.9%

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

      2. clear-num 57.9%

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

      3. un-div-inv 57.9%

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

   4. Applied egg-rr 57.9%

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

      1. *-commutative 57.9%

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

      2. unpow2 57.9%

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

   6. Applied egg-rr 57.9%

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

      1. associate-*r* 54.8%

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

   8. Simplified 54.8%

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

   9. Taylor expanded in t around inf 37.3%

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

       1. pow1/2 40.4%

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

       2. *-commutative 40.4%

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

       3. associate-*l* 40.4%

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

   11. Applied egg-rr 40.4%

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

   if 5.4999999999999996e204 < t

   1. Initial program 42.6%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in l around 0 50.2%

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

      1. associate-*r* 50.2%

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

   7. Simplified 50.2%

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

      1. pow1/2 65.2%

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

      2. associate-*l* 65.2%

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

   9. Applied egg-rr 65.2%

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

Alternative 14: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 3.7 \cdot 10^{-189}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 3.7e-189)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* t (* 2.0 (* 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 (U <= 3.7e-189) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((t * (2.0 * (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 (u <= 3.7d-189) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((t * (2.0d0 * (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 (U <= 3.7e-189) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 3.7e-189:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 3.7e-189)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(t * Float64(2.0 * 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 (U <= 3.7e-189)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((t * (2.0 * (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[U, 3.7e-189], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 3.70000000000000019e-189

   1. Initial program 49.7%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in l around 0 34.4%

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

      1. associate-*r* 34.5%

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

   7. Simplified 34.5%

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

      1. pow1/2 40.3%

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

      2. associate-*l* 40.3%

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

   9. Applied egg-rr 40.3%

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

   if 3.70000000000000019e-189 < U

   1. Initial program 57.4%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in l around 0 42.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.7 \cdot 10^{-189}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9.5 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 9.5e+84)
   (sqrt (* (* 2.0 U) (* n t)))
   (sqrt (* -2.0 (* U (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9.5e+84) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt((-2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 9.5d+84) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt(((-2.0d0) * (u * (n * t))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9.5e+84) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt((-2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 9.5e+84:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt((-2.0 * (U * (n * t))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 9.5e+84)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 9.5e+84)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt((-2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 9.5e+84], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.49999999999999979e84

   1. Initial program 57.6%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in l around 0 40.9%

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

      1. associate-*r* 40.9%

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

   7. Simplified 40.9%

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

   if 9.49999999999999979e84 < l

   1. Initial program 31.3%

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

   3. Taylor expanded in U around inf 6.4%

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

      1. unpow2 6.4%

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

      2. times-frac 6.6%

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

      3. unpow2 6.6%

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

   5. Simplified 6.6%

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

      1. add-sqr-sqrt 6.6%

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

      2. pow1/2 6.6%

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

      3. pow1/2 21.7%

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

      4. pow-prod-down 21.8%

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

   7. Applied egg-rr 20.4%

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

   8. Taylor expanded in t around -inf 16.1%

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

Alternative 16: 34.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.4 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.4e+86)
   (sqrt (* n (* t (* 2.0 U))))
   (sqrt (* -2.0 (* U (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.4e+86) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = sqrt((-2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.4d+86) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    else
        tmp = sqrt(((-2.0d0) * (u * (n * t))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.4e+86) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = Math.sqrt((-2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.4e+86:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	else:
		tmp = math.sqrt((-2.0 * (U * (n * t))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.4e+86)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.4e+86)
		tmp = sqrt((n * (t * (2.0 * U))));
	else
		tmp = sqrt((-2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.4e+86], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.40000000000000002e86

   1. Initial program 57.6%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in l around 0 40.9%

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

      1. associate-*r* 40.9%

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

   7. Simplified 40.9%

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

      1. *-commutative 40.9%

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

   9. Applied egg-rr 40.9%

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

       1. associate-*l* 38.2%

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

   11. Simplified 38.2%

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

   if 1.40000000000000002e86 < l

   1. Initial program 31.3%

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

   3. Taylor expanded in U around inf 6.4%

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

      1. unpow2 6.4%

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

      2. times-frac 6.6%

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

      3. unpow2 6.6%

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

   5. Simplified 6.6%

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

      1. add-sqr-sqrt 6.6%

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

      2. pow1/2 6.6%

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

      3. pow1/2 21.7%

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

      4. pow-prod-down 21.8%

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

   7. Applied egg-rr 20.4%

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

   8. Taylor expanded in t around -inf 16.1%

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

Alternative 17: 36.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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((t * (2.0d0 * (n * u))))
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((t * (2.0 * (n * U))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

3. Simplified 55.2%

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

5. Taylor expanded in l around 0 36.8%

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

6. Final simplification 36.8%

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

Alternative 18: 4.5% 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 52.7%

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

3. Taylor expanded in U around inf 34.7%

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

   1. unpow2 34.7%

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

   2. times-frac 33.4%

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

   3. unpow2 33.4%

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

5. Simplified 33.4%

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

   1. add-sqr-sqrt 33.4%

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

   2. pow1/2 33.4%

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

   3. pow1/2 42.3%

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

   4. pow-prod-down 29.3%

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

7. Applied egg-rr 31.3%

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

8. Taylor expanded in t around -inf 6.4%

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

Reproduce

herbie shell --seed 2024097 
(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*))))))