Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 57.9%

Time: 30.6s

Alternatives: 16

Speedup: 0.7×

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: 49.6% 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: 57.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\\ \mathbf{if}\;n \leq -1.65 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 \cdot \left(U* - U\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{l\_m} \cdot \frac{Om}{l\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, t\_1, \left(2 \cdot \frac{{l\_m}^{1.5}}{Om}\right) \cdot \sqrt{l\_m}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l_m Om) 2.0))))
   (if (<= n -1.65e-184)
     (sqrt
      (*
       (* 2.0 (* n U))
       (+ t (- (* t_1 (- U* U)) (* 2.0 (* l_m (/ l_m Om)))))))
     (if (<= n 1.85e-239)
       (sqrt
        (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (* (/ 1.0 l_m) (/ Om l_m)))))))))
       (*
        (sqrt (* n 2.0))
        (sqrt
         (*
          U
          (-
           t
           (fma
            (- U U*)
            t_1
            (* (* 2.0 (/ (pow l_m 1.5) Om)) (sqrt l_m)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * pow((l_m / Om), 2.0);
	double tmp;
	if (n <= -1.65e-184) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((t_1 * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	} else if (n <= 1.85e-239) {
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma((U - U_42_), t_1, ((2.0 * (pow(l_m, 1.5) / Om)) * sqrt(l_m))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	tmp = 0.0
	if (n <= -1.65e-184)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(t_1 * Float64(U_42_ - U)) - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	elseif (n <= 1.85e-239)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Float64(1.0 / l_m) * Float64(Om / l_m)))))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(Float64(U - U_42_), t_1, Float64(Float64(2.0 * Float64((l_m ^ 1.5) / Om)) * sqrt(l_m)))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.65e-184], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.85e-239], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(2.0 * N[(-1.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(U - U$42$), $MachinePrecision] * t$95$1 + N[(N[(2.0 * N[(N[Power[l$95$m, 1.5], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6499999999999999e-184

   1. Initial program 59.7%

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

   3. Simplified 60.7%

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

   if -1.6499999999999999e-184 < n < 1.85000000000000008e-239

   1. Initial program 45.7%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in n around 0 60.5%

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

      1. clear-num 60.5%

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

      2. inv-pow 60.5%

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

   7. Applied egg-rr 60.5%

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

      1. unpow-1 60.5%

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

   9. Simplified 60.5%

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

       1. *-un-lft-identity 60.5%

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

       2. unpow2 60.5%

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

       3. times-frac 62.9%

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

   11. Applied egg-rr 62.9%

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

   if 1.85000000000000008e-239 < n

   1. Initial program 51.2%

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

   3. Simplified 52.6%

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

      1. sqrt-prod 69.8%

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

      2. fma-undefine 69.8%

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

      3. associate-*r* 70.7%

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

      4. +-commutative 70.7%

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

      5. *-commutative 70.7%

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

      6. fma-define 70.7%

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

      7. associate-*r/ 67.2%

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

      8. pow2 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*r/ 67.2%

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

   8. Simplified 67.2%

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

      1. associate-/l* 67.2%

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

      2. unpow2 67.2%

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

      3. associate-*r/ 70.7%

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

      4. *-commutative 70.7%

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

      5. add-sqr-sqrt 25.2%

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

      6. associate-*l* 25.2%

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

      7. associate-*r* 25.2%

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

      8. associate-*l/ 23.5%

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

      9. pow1 23.5%

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

      10. pow1/2 23.5%

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

      11. pow-prod-up 23.5%

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

      12. metadata-eval 23.5%

          \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \left(2 \cdot \frac{{\ell}^{\color{blue}{1.5}}}{Om}\right) \cdot \sqrt{\ell}\right)\right)} \]

   10. Applied egg-rr 23.5%

       \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\left(2 \cdot \frac{{\ell}^{1.5}}{Om}\right) \cdot \sqrt{\ell}}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{\ell} \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \left(2 \cdot \frac{{\ell}^{1.5}}{Om}\right) \cdot \sqrt{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_3\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \left(n \cdot \left(t\_2 \cdot \left(U* - U\right)\right) - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \left(\sqrt{l\_m} \cdot \left(\frac{l\_m}{Om} \cdot \sqrt{l\_m}\right)\right)\right) + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(n \cdot {l\_m}^{2}\right) \cdot \left(\frac{2}{Om} - n \cdot \frac{U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (pow (/ l_m Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (* t_1 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_3))))
   (if (<= t_4 0.0)
     (*
      (sqrt (* n 2.0))
      (sqrt
       (* U (+ t (- (* n (* t_2 (- U* U))) (* 2.0 (/ (pow l_m 2.0) Om)))))))
     (if (<= t_4 INFINITY)
       (sqrt
        (* t_1 (+ (- t (* 2.0 (* (sqrt l_m) (* (/ l_m Om) (sqrt l_m))))) t_3)))
       (sqrt
        (*
         -2.0
         (*
          U
          (*
           (* n (pow l_m 2.0))
           (- (/ 2.0 Om) (* n (/ U* (pow Om 2.0))))))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[Sqrt[l$95$m], $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - N[(n * N[(U$42$ / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

      1. sqrt-prod 50.2%

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

      2. fma-undefine 50.2%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. *-commutative 52.8%

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

      6. fma-define 52.8%

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

      7. associate-*r/ 52.7%

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

      8. pow2 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r/ 52.7%

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

   8. Simplified 52.7%

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

      1. fma-undefine 52.7%

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

      2. *-commutative 52.7%

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

      3. associate-*l* 50.2%

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

      4. associate-/l* 50.2%

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

   10. Applied egg-rr 50.2%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

      1. associate-*r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. add-sqr-sqrt 32.2%

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

      4. associate-*r* 32.2%

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

   4. Applied egg-rr 32.2%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #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 0.0%

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

      1. associate-*r/ 0.2%

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

      2. *-commutative 0.2%

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

      3. add-sqr-sqrt 0.2%

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

      4. associate-*r* 0.2%

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

   4. Applied egg-rr 0.2%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-*r* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

      4. metadata-eval 32.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 32.0%

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

      11. metadata-eval 32.0%

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

      12. associate-/l* 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in U around 0 32.0%

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

      1. associate-*r/ 32.0%

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

      2. neg-mul-1 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 34.9%

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

Alternative 3: 61.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := t\_1 \cdot \left(U* - U\right)\\ t_3 := U \cdot \left(n \cdot 2\right)\\ t_4 := t\_3 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, t\_1, \frac{2 \cdot {l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(\left(t - 2 \cdot \left(\sqrt{l\_m} \cdot \left(\frac{l\_m}{Om} \cdot \sqrt{l\_m}\right)\right)\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(n \cdot {l\_m}^{2}\right) \cdot \left(\frac{2}{Om} - n \cdot \frac{U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l_m Om) 2.0)))
        (t_2 (* t_1 (- U* U)))
        (t_3 (* U (* n 2.0)))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2))))
   (if (<= t_4 0.0)
     (*
      (sqrt (* n 2.0))
      (sqrt (* U (- t (fma (- U U*) t_1 (/ (* 2.0 (pow l_m 2.0)) Om))))))
     (if (<= t_4 INFINITY)
       (sqrt
        (* t_3 (+ (- t (* 2.0 (* (sqrt l_m) (* (/ l_m Om) (sqrt l_m))))) t_2)))
       (sqrt
        (*
         -2.0
         (*
          U
          (*
           (* n (pow l_m 2.0))
           (- (/ 2.0 Om) (* n (/ U* (pow Om 2.0))))))))))))

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(U - U$42$), $MachinePrecision] * t$95$1 + N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[Sqrt[l$95$m], $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - N[(n * N[(U$42$ / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

      1. sqrt-prod 50.2%

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

      2. fma-undefine 50.2%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. *-commutative 52.8%

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

      6. fma-define 52.8%

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

      7. associate-*r/ 52.7%

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

      8. pow2 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r/ 52.7%

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

   8. Simplified 52.7%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

      1. associate-*r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. add-sqr-sqrt 32.2%

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

      4. associate-*r* 32.2%

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

   4. Applied egg-rr 32.2%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #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 0.0%

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

      1. associate-*r/ 0.2%

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

      2. *-commutative 0.2%

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

      3. add-sqr-sqrt 0.2%

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

      4. associate-*r* 0.2%

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

   4. Applied egg-rr 0.2%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-*r* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

      4. metadata-eval 32.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 32.0%

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

      11. metadata-eval 32.0%

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

      12. associate-/l* 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in U around 0 32.0%

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

      1. associate-*r/ 32.0%

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

      2. neg-mul-1 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 35.2%

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

Alternative 4: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left({l\_m}^{2} \cdot \left(n \cdot \left(\frac{2}{Om} - U* \cdot \frac{n}{{Om}^{2}}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

      1. sqrt-prod 50.2%

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

      2. fma-undefine 50.2%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. *-commutative 52.8%

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

      6. fma-define 52.8%

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

      7. associate-*r/ 52.7%

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

      8. pow2 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r/ 52.7%

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

   8. Simplified 52.7%

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

   9. Taylor expanded in n around 0 47.9%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

   3. Simplified 71.4%

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

   if +inf.0 < (*.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 0.0%

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

      1. associate-*r/ 0.2%

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

      2. *-commutative 0.2%

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

      3. add-sqr-sqrt 0.2%

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

      4. associate-*r* 0.2%

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

   4. Applied egg-rr 0.2%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-*r* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

      4. metadata-eval 32.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 32.0%

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

      11. metadata-eval 32.0%

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

      12. associate-/l* 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in U around 0 31.9%

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

      1. +-commutative 31.9%

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

      2. mul-1-neg 31.9%

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

      3. unsub-neg 31.9%

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

      4. associate-*r/ 31.9%

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

      5. metadata-eval 31.9%

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

      6. associate-/l* 31.8%

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

   10. Simplified 31.8%

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

4. Final simplification 63.6%

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

Alternative 5: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(n \cdot {l\_m}^{2}\right) \cdot \left(\frac{2}{Om} - n \cdot \frac{U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

      1. sqrt-prod 50.2%

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

      2. fma-undefine 50.2%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. *-commutative 52.8%

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

      6. fma-define 52.8%

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

      7. associate-*r/ 52.7%

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

      8. pow2 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r/ 52.7%

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

   8. Simplified 52.7%

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

   9. Taylor expanded in n around 0 47.9%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

   3. Simplified 71.4%

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

   if +inf.0 < (*.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 0.0%

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

      1. associate-*r/ 0.2%

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

      2. *-commutative 0.2%

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

      3. add-sqr-sqrt 0.2%

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

      4. associate-*r* 0.2%

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

   4. Applied egg-rr 0.2%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-*r* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

      4. metadata-eval 32.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 32.0%

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

      11. metadata-eval 32.0%

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

      12. associate-/l* 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in U around 0 32.0%

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

      1. associate-*r/ 32.0%

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

      2. neg-mul-1 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 63.6%

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

Alternative 6: 61.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\\ t_3 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \left(n \cdot \left(t\_1 \cdot \left(U* - U\right)\right) - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(n \cdot {l\_m}^{2}\right) \cdot \left(\frac{2}{Om} - n \cdot \frac{U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

      1. sqrt-prod 50.2%

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

      2. fma-undefine 50.2%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. *-commutative 52.8%

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

      6. fma-define 52.8%

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

      7. associate-*r/ 52.7%

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

      8. pow2 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r/ 52.7%

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

   8. Simplified 52.7%

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

      1. fma-undefine 52.7%

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

      2. *-commutative 52.7%

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

      3. associate-*l* 50.2%

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

      4. associate-/l* 50.2%

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

   10. Applied egg-rr 50.2%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

   3. Simplified 71.4%

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

   if +inf.0 < (*.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 0.0%

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

      1. associate-*r/ 0.2%

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

      2. *-commutative 0.2%

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

      3. add-sqr-sqrt 0.2%

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

      4. associate-*r* 0.2%

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

   4. Applied egg-rr 0.2%

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

   5. Taylor expanded in l around inf 31.9%

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

      1. associate-*r* 32.0%

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

      2. associate-*r/ 32.0%

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

      3. metadata-eval 32.0%

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

      4. metadata-eval 32.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 32.0%

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

      11. metadata-eval 32.0%

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

      12. associate-/l* 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in U around 0 32.0%

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

      1. associate-*r/ 32.0%

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

      2. neg-mul-1 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 63.9%

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

Alternative 7: 60.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{l\_m} \cdot \frac{Om}{l\_m}}\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U*} \cdot \sqrt{U}\right) \cdot \left(l\_m \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* U (* n 2.0)) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_2 0.0)
     (sqrt
      (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (* (/ 1.0 l_m) (/ Om l_m)))))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (* (* (sqrt U*) (sqrt U)) (* l_m (/ (* n (sqrt 2.0)) Om)))))))

C

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (Math.sqrt(U_42_) * Math.sqrt(U)) * (l_m * ((n * Math.sqrt(2.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = (math.sqrt(U_42_) * math.sqrt(U)) * (l_m * ((n * math.sqrt(2.0)) / Om))
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (sqrt(U_42_) * sqrt(U)) * (l_m * ((n * sqrt(2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(2.0 * N[(-1.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[U$42$], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in n around 0 41.3%

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

      1. clear-num 41.3%

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

      2. inv-pow 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. unpow-1 41.3%

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

   9. Simplified 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. unpow2 41.3%

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

       3. times-frac 43.9%

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

   11. Applied egg-rr 43.9%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

   3. Simplified 71.4%

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

   if +inf.0 < (*.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 0.0%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in U* around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. distribute-rgt-neg-in 0.0%

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

      4. associate-/l* 0.0%

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

      5. associate-*r* 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 22.3%

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

      9. neg-mul-1 22.3%

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

   7. Simplified 22.3%

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

      1. pow1/2 22.3%

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

      2. *-commutative 22.3%

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

      3. unpow-prod-down 11.4%

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

      4. pow1/2 11.4%

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

      5. pow1/2 11.4%

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

   9. Applied egg-rr 11.4%

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

4. Final simplification 60.7%

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

Alternative 8: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{l\_m} \cdot \frac{Om}{l\_m}}\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* U (* n 2.0)) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_2 0.0)
     (sqrt
      (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (* (/ 1.0 l_m) (/ Om l_m)))))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l_m 2.0)) 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 t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l_m, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Float64(1.0 / l_m) * Float64(Om / l_m)))))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))) ^ 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 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(2.0 * N[(-1.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 0.0

   1. Initial program 13.6%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in n around 0 41.3%

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

      1. clear-num 41.3%

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

      2. inv-pow 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. unpow-1 41.3%

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

   9. Simplified 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. unpow2 41.3%

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

       3. times-frac 43.9%

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

   11. Applied egg-rr 43.9%

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

   if 0.0 < (*.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*)))) < +inf.0

   1. Initial program 69.9%

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

   3. Simplified 71.4%

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

   if +inf.0 < (*.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 0.0%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in n around 0 4.6%

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

      1. pow1/2 28.9%

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

      2. associate-*r* 24.8%

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

      3. *-commutative 24.8%

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

      4. associate-*r/ 24.8%

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

   7. Applied egg-rr 24.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{\ell} \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -2.3 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -2.3e-148)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+
      t
      (- (* (* n (pow (/ l_m Om) 2.0)) (- U* U)) (* 2.0 (* l_m (/ l_m Om)))))))
   (if (<= U -5e-310)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l_m 2.0) Om)))))))
     (* (sqrt (* 2.0 U)) (sqrt (* n (- t (/ (* 2.0 (pow l_m 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 (U <= -2.3e-148) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (((n * pow((l_m / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	} else if (U <= -5e-310) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * (t - ((2.0 * pow(l_m, 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 (u <= (-2.3d-148)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)) - (2.0d0 * (l_m * (l_m / om)))))))
    else if (u <= (-5d-310)) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * (t - ((2.0d0 * (l_m ** 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 (U <= -2.3e-148) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	} else if (U <= -5e-310) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -2.3e-148:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))))
	elif U <= -5e-310:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * (t - ((2.0 * math.pow(l_m, 2.0)) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -2.3e-148)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)) - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	elseif (U <= -5e-310)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * Float64(t - Float64(Float64(2.0 * (l_m ^ 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 (U <= -2.3e-148)
		tmp = sqrt(((2.0 * (n * U)) * (t + (((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	elseif (U <= -5e-310)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l_m ^ 2.0) / Om)))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * (t - ((2.0 * (l_m ^ 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[U, -2.3e-148], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, -5e-310], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -2.29999999999999997e-148

   1. Initial program 66.5%

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

   3. Simplified 70.1%

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

   if -2.29999999999999997e-148 < U < -4.999999999999985e-310

   1. Initial program 35.4%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in n around 0 48.3%

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

   if -4.999999999999985e-310 < U

   1. Initial program 51.2%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in n around 0 51.3%

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

      1. pow1/2 53.3%

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

      2. associate-*r* 53.3%

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

      3. unpow-prod-down 64.9%

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

      4. pow1/2 63.0%

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

      5. associate-*r/ 63.0%

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

   7. Applied egg-rr 63.0%

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

      1. unpow1/2 63.0%

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

   9. Simplified 63.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.3 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{Om}{{l\_m}^{2}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -8.2e-110)
   (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l_m 2.0)) Om)))) 0.5)
   (if (<= U 2.65e+141)
     (sqrt (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (/ Om (pow l_m 2.0)))))))))
     (* (sqrt (* 2.0 U)) (sqrt (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -8.2e-110) {
		tmp = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l_m, 2.0)) / Om)))), 0.5);
	} else if (U <= 2.65e+141) {
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / (Om / pow(l_m, 2.0)))))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-8.2d-110)) then
        tmp = (2.0d0 * ((n * u) * (t - ((2.0d0 * (l_m ** 2.0d0)) / om)))) ** 0.5d0
    else if (u <= 2.65d+141) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (2.0d0 * ((-1.0d0) / (om / (l_m ** 2.0d0)))))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -8.2e-110) {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))), 0.5);
	} else if (U <= 2.65e+141) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / (Om / Math.pow(l_m, 2.0)))))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -8.2e-110:
		tmp = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))), 0.5)
	elif U <= 2.65e+141:
		tmp = math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / (Om / math.pow(l_m, 2.0)))))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -8.2e-110)
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5;
	elseif (U <= 2.65e+141)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Om / (l_m ^ 2.0)))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -8.2e-110)
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5;
	elseif (U <= 2.65e+141)
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / (Om / (l_m ^ 2.0)))))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -8.2e-110], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 2.65e+141], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(2.0 * N[(-1.0 / N[(Om / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -8.19999999999999965e-110

   1. Initial program 65.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in n around 0 54.1%

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

      1. pow1/2 58.6%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*r/ 63.5%

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

   7. Applied egg-rr 63.5%

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

   if -8.19999999999999965e-110 < U < 2.65e141

   1. Initial program 49.0%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in n around 0 49.5%

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

      1. clear-num 49.6%

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

      2. inv-pow 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. unpow-1 49.6%

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

   9. Simplified 49.6%

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

   if 2.65e141 < U

   1. Initial program 52.2%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in t around inf 56.9%

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

      1. pow1/2 61.5%

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

      2. associate-*r* 61.5%

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

      3. unpow-prod-down 78.1%

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

      4. pow1/2 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. unpow1/2 78.1%

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

   9. Simplified 78.1%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{Om}{{\ell}^{2}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 1.12 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{l\_m} \cdot \frac{Om}{l\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -8.2e-110)
   (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l_m 2.0)) Om)))) 0.5)
   (if (<= U 1.12e+141)
     (sqrt
      (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (* (/ 1.0 l_m) (/ Om l_m)))))))))
     (* (sqrt (* 2.0 U)) (sqrt (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -8.2e-110) {
		tmp = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l_m, 2.0)) / Om)))), 0.5);
	} else if (U <= 1.12e+141) {
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-8.2d-110)) then
        tmp = (2.0d0 * ((n * u) * (t - ((2.0d0 * (l_m ** 2.0d0)) / om)))) ** 0.5d0
    else if (u <= 1.12d+141) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (2.0d0 * ((-1.0d0) / ((1.0d0 / l_m) * (om / l_m)))))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -8.2e-110) {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))), 0.5);
	} else if (U <= 1.12e+141) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -8.2e-110:
		tmp = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))), 0.5)
	elif U <= 1.12e+141:
		tmp = math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -8.2e-110)
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5;
	elseif (U <= 1.12e+141)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Float64(1.0 / l_m) * Float64(Om / l_m)))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -8.2e-110)
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5;
	elseif (U <= 1.12e+141)
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -8.2e-110], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 1.12e+141], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(2.0 * N[(-1.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -8.19999999999999965e-110

   1. Initial program 65.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in n around 0 54.1%

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

      1. pow1/2 58.6%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*r/ 63.5%

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

   7. Applied egg-rr 63.5%

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

   if -8.19999999999999965e-110 < U < 1.11999999999999993e141

   1. Initial program 49.0%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in n around 0 49.5%

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

      1. clear-num 49.6%

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

      2. inv-pow 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. unpow-1 49.6%

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

   9. Simplified 49.6%

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

       1. *-un-lft-identity 49.6%

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

       2. unpow2 49.6%

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

       3. times-frac 49.6%

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

   11. Applied egg-rr 49.6%

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

   if 1.11999999999999993e141 < U

   1. Initial program 52.2%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in t around inf 56.9%

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

      1. pow1/2 61.5%

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

      2. associate-*r* 61.5%

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

      3. unpow-prod-down 78.1%

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

      4. pow1/2 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. unpow1/2 78.1%

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

   9. Simplified 78.1%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 1.12 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{\ell} \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{l\_m} \cdot \frac{Om}{l\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 1.8e+141)
   (sqrt (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (* (/ 1.0 l_m) (/ Om l_m)))))))))
   (* (sqrt (* 2.0 U)) (sqrt (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 1.8e+141) {
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 1.8d+141) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (2.0d0 * ((-1.0d0) / ((1.0d0 / l_m) * (om / l_m)))))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 1.8e+141) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 1.8e+141:
		tmp = math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 1.8e+141)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Float64(1.0 / l_m) * Float64(Om / l_m)))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.8e+141)
		tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, 1.8e+141], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(2.0 * N[(-1.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.8000000000000001e141

   1. Initial program 54.0%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in n around 0 51.0%

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

      1. clear-num 51.0%

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

      2. inv-pow 51.0%

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

   7. Applied egg-rr 51.0%

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

      1. unpow-1 51.0%

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

   9. Simplified 51.0%

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

       1. *-un-lft-identity 51.0%

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

       2. unpow2 51.0%

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

       3. times-frac 52.2%

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

   11. Applied egg-rr 52.2%

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

   if 1.8000000000000001e141 < U

   1. Initial program 52.2%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in t around inf 56.9%

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

      1. pow1/2 61.5%

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

      2. associate-*r* 61.5%

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

      3. unpow-prod-down 78.1%

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

      4. pow1/2 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. unpow1/2 78.1%

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

   9. Simplified 78.1%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{\ell} \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{l\_m} \cdot \frac{Om}{l\_m}}\right)\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (* 2.0 (* U (* n (+ t (* 2.0 (/ -1.0 (* (/ 1.0 l_m) (/ Om l_m))))))))))

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 + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
}

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 + (2.0d0 * ((-1.0d0) / ((1.0d0 / l_m) * (om / l_m)))))))))
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 + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Float64(1.0 / l_m) * Float64(Om / l_m)))))))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * (t + (2.0 * (-1.0 / ((1.0 / l_m) * (Om / l_m)))))))));
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 * N[(t + N[(2.0 * N[(-1.0 / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.9%

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

3. Simplified 53.4%

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

5. Taylor expanded in n around 0 51.5%

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

   1. clear-num 51.5%

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

   2. inv-pow 51.5%

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

7. Applied egg-rr 51.5%

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

   1. unpow-1 51.5%

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

9. Simplified 51.5%

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

    1. *-un-lft-identity 51.5%

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

    2. unpow2 51.5%

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

    3. times-frac 52.7%

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

11. Applied egg-rr 52.7%

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

12. Final simplification 52.7%

    \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + 2 \cdot \frac{-1}{\frac{1}{\ell} \cdot \frac{Om}{\ell}}\right)\right)\right)} \]
13. Add Preprocessing

Alternative 14: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -6.5 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -6.5e-121)
   (sqrt (* 2.0 (* (* n U) 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 (U <= -6.5e-121) {
		tmp = sqrt((2.0 * ((n * U) * 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 (u <= (-6.5d-121)) then
        tmp = sqrt((2.0d0 * ((n * u) * 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 (U <= -6.5e-121) {
		tmp = Math.sqrt((2.0 * ((n * U) * 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 U <= -6.5e-121:
		tmp = math.sqrt((2.0 * ((n * U) * 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 (U <= -6.5e-121)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * 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 (U <= -6.5e-121)
		tmp = sqrt((2.0 * ((n * U) * 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[U, -6.5e-121], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * 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 U < -6.5000000000000003e-121

   1. Initial program 66.3%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in t around inf 42.1%

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

      1. associate-*r* 48.3%

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

   7. Simplified 48.3%

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

   if -6.5000000000000003e-121 < U

   1. Initial program 48.5%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 40.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -6.5 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.4% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* 2.0 (* 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_) {
	return pow((2.0 * (U * (n * t))), 0.5);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Simplified 53.4%

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

5. Taylor expanded in t around inf 40.9%

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

   1. pow1/2 43.8%

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

7. Applied egg-rr 43.8%

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

8. Final simplification 43.8%

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

Alternative 16: 35.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Simplified 53.4%

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

5. Taylor expanded in t around inf 40.9%

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

6. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2024079 
(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*))))))