Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 62.6%

Time: 27.9s

Alternatives: 22

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{U* - U}{Om}\\ \mathbf{if}\;\ell \leq 58000000:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot t_1 + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, t_1, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (- U* U) Om)))
   (if (<= l 58000000.0)
     (sqrt
      (*
       2.0
       (*
        n
        (*
         U
         (fma
          n
          (* (pow (/ l Om) 2.0) (- U* U))
          (fma l (* (/ l Om) -2.0) t))))))
     (if (<= l 6e+149)
       (sqrt
        (*
         2.0
         (* U (fma n t (* (* l l) (* n (+ (* (/ n Om) t_1) (/ -2.0 Om))))))))
       (*
        (sqrt (* (* n U) (fma (/ n Om) t_1 (/ -2.0 Om))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (U_42_ - U) / Om;
	double tmp;
	if (l <= 58000000.0) {
		tmp = sqrt((2.0 * (n * (U * fma(n, (pow((l / Om), 2.0) * (U_42_ - U)), fma(l, ((l / Om) * -2.0), t))))));
	} else if (l <= 6e+149) {
		tmp = sqrt((2.0 * (U * fma(n, t, ((l * l) * (n * (((n / Om) * t_1) + (-2.0 / Om))))))));
	} else {
		tmp = sqrt(((n * U) * fma((n / Om), t_1, (-2.0 / Om)))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(U_42_ - U) / Om)
	tmp = 0.0
	if (l <= 58000000.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * fma(n, Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U)), fma(l, Float64(Float64(l / Om) * -2.0), t))))));
	elseif (l <= 6e+149)
		tmp = sqrt(Float64(2.0 * Float64(U * fma(n, t, Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n / Om) * t_1) + Float64(-2.0 / Om))))))));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * fma(Float64(n / Om), t_1, Float64(-2.0 / Om)))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, 58000000.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+149], N[Sqrt[N[(2.0 * N[(U * N[(n * t + N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * t$95$1 + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.8e7

   1. Initial program 55.1%

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

   3. Simplified 58.0%

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

      1. pow1 58.0%

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

   5. Applied egg-rr 58.0%

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

      1. unpow1 58.0%

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

      2. associate-*l* 59.9%

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

      3. *-commutative 59.9%

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

      4. fma-udef 59.9%

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

      5. associate-*l* 59.9%

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

      6. fma-def 59.9%

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

   7. Simplified 59.9%

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

   if 5.8e7 < l < 6.00000000000000007e149

   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 61.1%

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

   4. Taylor expanded in l around 0 73.4%

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

      1. distribute-lft-out 73.4%

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

      2. fma-def 73.4%

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

      3. unpow2 73.4%

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

      4. sub-neg 73.4%

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

      5. unpow2 73.4%

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

      6. times-frac 77.3%

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

      7. associate-*r/ 77.3%

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

      8. metadata-eval 77.3%

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

      9. distribute-neg-frac 77.3%

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

      10. metadata-eval 77.3%

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

   6. Simplified 77.3%

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

   if 6.00000000000000007e149 < l

   1. Initial program 11.0%

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

   3. Simplified 28.4%

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

      1. pow1 28.4%

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

   5. Applied egg-rr 28.4%

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

      1. unpow1 28.4%

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

      2. associate-*l* 30.2%

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

      3. *-commutative 30.2%

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

      4. fma-udef 30.2%

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

      5. associate-*l* 30.2%

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

      6. fma-def 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in l around inf 60.9%

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

      1. associate-*r* 70.5%

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

      2. *-commutative 70.5%

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

      3. sub-neg 70.5%

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

      4. unpow2 70.5%

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

      5. times-frac 74.6%

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

      6. associate-*r/ 74.6%

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

      7. metadata-eval 74.6%

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

      8. distribute-neg-frac 74.6%

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

      9. metadata-eval 74.6%

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

      10. fma-udef 74.6%

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

   10. Simplified 74.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 58000000:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := t_1 \cdot \left(t_2 - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_3 \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 10^{+304}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* t_1 (- t_2 (- (* 2.0 (/ (* l l) Om)) t)))))
   (if (<= t_3 5e-279)
     (sqrt (* 2.0 (* n (* U (+ t (* -2.0 (/ l (/ Om l))))))))
     (if (<= t_3 1e+304)
       (sqrt (* t_1 (+ (- t (* 2.0 (* l (/ l Om)))) t_2)))
       (*
        (sqrt (* (* n U) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = t_1 * (t_2 - ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_3 <= 5e-279) {
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	} else if (t_3 <= 1e+304) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_2)));
	} else {
		tmp = sqrt(((n * U) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om)))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)))
	tmp = 0.0
	if (t_3 <= 5e-279)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))))));
	elseif (t_3 <= 1e+304)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_2)));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-279], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+304], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.99999999999999969e-279

   1. Initial program 23.6%

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

   3. Simplified 25.7%

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

      1. pow1 25.7%

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

   5. Applied egg-rr 25.7%

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

      1. unpow1 25.7%

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

      2. associate-*l* 41.1%

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

      3. *-commutative 41.1%

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

      4. fma-udef 41.1%

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

      5. associate-*l* 41.1%

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

      6. fma-def 41.1%

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

   7. Simplified 41.1%

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

   8. Taylor expanded in n around 0 40.8%

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

      1. unpow2 40.8%

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

      2. associate-/l* 42.9%

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

   10. Simplified 42.9%

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

   if 4.99999999999999969e-279 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 9.9999999999999994e303

   1. Initial program 98.6%

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

      1. associate-*l/ 98.6%

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

   3. Applied egg-rr 98.6%

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

   if 9.9999999999999994e303 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 22.0%

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

   3. Simplified 31.0%

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

      1. pow1 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. unpow1 31.0%

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

      2. associate-*l* 33.3%

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

      3. *-commutative 33.3%

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

      4. fma-udef 33.3%

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

      5. associate-*l* 33.3%

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

      6. fma-def 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in l around inf 23.5%

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

      1. associate-*r* 26.1%

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

      2. *-commutative 26.1%

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

      3. sub-neg 26.1%

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

      4. unpow2 26.1%

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

      5. times-frac 27.3%

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

      6. associate-*r/ 27.3%

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

      7. metadata-eval 27.3%

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

      8. distribute-neg-frac 27.3%

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

      9. metadata-eval 27.3%

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

      10. fma-udef 27.3%

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

   10. Simplified 27.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 10^{+304}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 3: 65.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := t_1 \cdot \left(t_2 - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_3 \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \frac{\ell}{\frac{Om}{\ell}} \cdot \frac{U*}{Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 10^{+304}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* t_1 (- t_2 (- (* 2.0 (/ (* l l) Om)) t)))))
   (if (<= t_3 5e-235)
     (sqrt
      (*
       2.0
       (*
        n
        (*
         U
         (fma n (* (/ l (/ Om l)) (/ U* Om)) (fma l (* (/ l Om) -2.0) t))))))
     (if (<= t_3 1e+304)
       (sqrt (* t_1 (+ (- t (* 2.0 (* l (/ l Om)))) t_2)))
       (*
        (sqrt (* (* n U) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = t_1 * (t_2 - ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_3 <= 5e-235) {
		tmp = sqrt((2.0 * (n * (U * fma(n, ((l / (Om / l)) * (U_42_ / Om)), fma(l, ((l / Om) * -2.0), t))))));
	} else if (t_3 <= 1e+304) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_2)));
	} else {
		tmp = sqrt(((n * U) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om)))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)))
	tmp = 0.0
	if (t_3 <= 5e-235)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * fma(n, Float64(Float64(l / Float64(Om / l)) * Float64(U_42_ / Om)), fma(l, Float64(Float64(l / Om) * -2.0), t))))));
	elseif (t_3 <= 1e+304)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_2)));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-235], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(n * N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+304], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.9999999999999998e-235

   1. Initial program 28.0%

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

   3. Simplified 30.0%

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

      1. pow1 30.0%

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

   5. Applied egg-rr 30.0%

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

      1. unpow1 30.0%

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

      2. associate-*l* 44.5%

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

      3. *-commutative 44.5%

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

      4. fma-udef 44.5%

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

      5. associate-*l* 44.5%

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

      6. fma-def 44.5%

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

   7. Simplified 44.5%

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

   8. Taylor expanded in U* around inf 36.3%

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

      1. *-commutative 36.3%

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

      2. unpow2 36.3%

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

      3. times-frac 44.4%

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

      4. unpow2 44.4%

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

      5. associate-/l* 46.4%

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

   10. Simplified 46.4%

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

   if 4.9999999999999998e-235 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 9.9999999999999994e303

   1. Initial program 98.5%

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

      1. associate-*l/ 98.6%

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

   3. Applied egg-rr 98.6%

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

   if 9.9999999999999994e303 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 22.0%

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

   3. Simplified 31.0%

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

      1. pow1 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. unpow1 31.0%

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

      2. associate-*l* 33.3%

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

      3. *-commutative 33.3%

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

      4. fma-udef 33.3%

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

      5. associate-*l* 33.3%

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

      6. fma-def 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in l around inf 23.5%

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

      1. associate-*r* 26.1%

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

      2. *-commutative 26.1%

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

      3. sub-neg 26.1%

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

      4. unpow2 26.1%

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

      5. times-frac 27.3%

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

      6. associate-*r/ 27.3%

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

      7. metadata-eval 27.3%

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

      8. distribute-neg-frac 27.3%

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

      9. metadata-eval 27.3%

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

      10. fma-udef 27.3%

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

   10. Simplified 27.3%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \frac{\ell}{\frac{Om}{\ell}} \cdot \frac{U*}{Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 10^{+304}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 4: 66.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := t_1 \cdot \left(t_2 - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right)\\ \mathbf{if}\;t_3 \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 10^{+304}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* t_1 (- t_2 (- (* 2.0 (/ (* l l) Om)) t)))))
   (if (<= t_3 5e-279)
     (sqrt (* 2.0 (* n (* U (+ t (* -2.0 (/ l (/ Om l))))))))
     (if (<= t_3 1e+304)
       (sqrt (* t_1 (+ (- t (* 2.0 (* l (/ l Om)))) t_2)))
       (*
        (* l (sqrt 2.0))
        (sqrt (* U (* n (+ (/ -2.0 Om) (* (/ n Om) (/ U* Om)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = t_1 * (t_2 - ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_3 <= 5e-279) {
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	} else if (t_3 <= 1e+304) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_2)));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = (n * ((l / om) ** 2.0d0)) * (u_42 - u)
    t_3 = t_1 * (t_2 - ((2.0d0 * ((l * l) / om)) - t))
    if (t_3 <= 5d-279) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((-2.0d0) * (l / (om / l))))))))
    else if (t_3 <= 1d+304) then
        tmp = sqrt((t_1 * ((t - (2.0d0 * (l * (l / om)))) + t_2)))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((u * (n * (((-2.0d0) / om) + ((n / om) * (u_42 / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = t_1 * (t_2 - ((2.0 * ((l * l) / Om)) - t));
	double tmp;
	if (t_3 <= 5e-279) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	} else if (t_3 <= 1e+304) {
		tmp = Math.sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_2)));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = t_1 * (t_2 - ((2.0 * ((l * l) / Om)) - t))
	tmp = 0
	if t_3 <= 5e-279:
		tmp = math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))))
	elif t_3 <= 1e+304:
		tmp = math.sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_2)))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)))
	tmp = 0.0
	if (t_3 <= 5e-279)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))))));
	elseif (t_3 <= 1e+304)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_2)));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (2.0 * n);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = t_1 * (t_2 - ((2.0 * ((l * l) / Om)) - t));
	tmp = 0.0;
	if (t_3 <= 5e-279)
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	elseif (t_3 <= 1e+304)
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_2)));
	else
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-279], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+304], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.99999999999999969e-279

   1. Initial program 23.6%

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

   3. Simplified 25.7%

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

      1. pow1 25.7%

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

   5. Applied egg-rr 25.7%

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

      1. unpow1 25.7%

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

      2. associate-*l* 41.1%

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

      3. *-commutative 41.1%

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

      4. fma-udef 41.1%

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

      5. associate-*l* 41.1%

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

      6. fma-def 41.1%

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

   7. Simplified 41.1%

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

   8. Taylor expanded in n around 0 40.8%

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

      1. unpow2 40.8%

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

      2. associate-/l* 42.9%

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

   10. Simplified 42.9%

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

   if 4.99999999999999969e-279 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 9.9999999999999994e303

   1. Initial program 98.6%

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

      1. associate-*l/ 98.6%

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

   3. Applied egg-rr 98.6%

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

   if 9.9999999999999994e303 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 22.0%

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

   3. Simplified 31.0%

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

      1. pow1 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. unpow1 31.0%

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

      2. associate-*l* 33.3%

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

      3. *-commutative 33.3%

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

      4. fma-udef 33.3%

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

      5. associate-*l* 33.3%

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

      6. fma-def 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in U* around inf 22.2%

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

      1. *-commutative 22.2%

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

      2. unpow2 22.2%

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

      3. times-frac 24.5%

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

      4. unpow2 24.5%

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

      5. associate-/l* 33.4%

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

   10. Simplified 33.4%

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

   11. Taylor expanded in l around inf 23.5%

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

       1. sub-neg 23.5%

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

       2. unpow2 23.5%

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

       3. times-frac 25.5%

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

       4. associate-*r/ 25.5%

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

       5. metadata-eval 25.5%

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

       6. distribute-neg-frac 25.5%

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

       7. metadata-eval 25.5%

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

   13. Simplified 25.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \leq 10^{+304}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 5: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right) + \frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot \frac{\frac{U*}{Om}}{Om} - \frac{n}{Om} \cdot \frac{U}{Om}\right) - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.35e-183)
   (* (sqrt 2.0) (sqrt (* t (* n U))))
   (if (<= l 1.02e-47)
     (sqrt
      (*
       (* 2.0 U)
       (*
        n
        (+ (fma l (* (/ l Om) -2.0) t) (* (/ U* Om) (/ (* n (* l l)) Om))))))
     (if (<= l 2.25e-23)
       (sqrt
        (*
         2.0
         (*
          n
          (*
           U
           (+
            t
            (*
             (* l l)
             (-
              (- (* n (/ (/ U* Om) Om)) (* (/ n Om) (/ U Om)))
              (/ 2.0 Om))))))))
       (if (<= l 1.9e+151)
         (sqrt
          (*
           2.0
           (*
            U
            (fma
             n
             t
             (* (* l l) (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om))))))))
         (*
          (* l (sqrt 2.0))
          (sqrt (* U (* n (+ (/ -2.0 Om) (* (/ n Om) (/ U* Om))))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.35e-183) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else if (l <= 1.02e-47) {
		tmp = sqrt(((2.0 * U) * (n * (fma(l, ((l / Om) * -2.0), t) + ((U_42_ / Om) * ((n * (l * l)) / Om))))));
	} else if (l <= 2.25e-23) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * ((U_42_ / Om) / Om)) - ((n / Om) * (U / Om))) - (2.0 / Om))))))));
	} else if (l <= 1.9e+151) {
		tmp = sqrt((2.0 * (U * fma(n, t, ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.35e-183)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	elseif (l <= 1.02e-47)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(fma(l, Float64(Float64(l / Om) * -2.0), t) + Float64(Float64(U_42_ / Om) * Float64(Float64(n * Float64(l * l)) / Om))))));
	elseif (l <= 2.25e-23)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(Float64(U_42_ / Om) / Om)) - Float64(Float64(n / Om) * Float64(U / Om))) - Float64(2.0 / Om))))))));
	elseif (l <= 1.9e+151)
		tmp = sqrt(Float64(2.0 * Float64(U * fma(n, t, Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.35e-183], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.02e-47], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision] + N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.25e-23], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(N[(U$42$ / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.9e+151], N[Sqrt[N[(2.0 * N[(U * N[(n * t + N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 1.35000000000000004e-183

   1. Initial program 53.0%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-*r* 39.3%

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

      3. *-commutative 39.3%

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

   6. Simplified 39.3%

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

   if 1.35000000000000004e-183 < l < 1.02000000000000002e-47

   1. Initial program 62.0%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in U around 0 54.3%

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

   6. Simplified 58.3%

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

   if 1.02000000000000002e-47 < l < 2.24999999999999987e-23

   1. Initial program 52.0%

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

   3. Simplified 52.0%

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

      1. pow1 52.0%

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

   5. Applied egg-rr 52.0%

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

      1. unpow1 52.0%

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

      2. associate-*l* 76.1%

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

      3. *-commutative 76.1%

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

      4. fma-udef 76.1%

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

      5. associate-*l* 76.1%

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

      6. fma-def 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in l around 0 76.1%

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

      1. unpow2 76.1%

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

      2. associate-/l* 76.1%

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

      3. unpow2 76.1%

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

      4. associate-*r/ 76.1%

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

      5. metadata-eval 76.1%

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

   10. Simplified 76.1%

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

   11. Taylor expanded in U* around 0 74.6%

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

       1. +-commutative 74.6%

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

       2. mul-1-neg 74.6%

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

       3. unsub-neg 74.6%

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

       4. associate-/l* 74.6%

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

       5. associate-/r/ 74.6%

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

       6. unpow2 74.6%

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

       7. associate-/r* 74.6%

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

       8. unpow2 74.6%

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

       9. times-frac 74.6%

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

   13. Simplified 74.6%

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

   if 2.24999999999999987e-23 < l < 1.9e151

   1. Initial program 68.7%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in l around 0 75.4%

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

      1. distribute-lft-out 75.4%

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

      2. fma-def 75.4%

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

      3. unpow2 75.4%

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

      4. sub-neg 75.4%

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

      5. unpow2 75.4%

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

      6. times-frac 78.6%

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

      7. associate-*r/ 78.6%

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

      8. metadata-eval 78.6%

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

      9. distribute-neg-frac 78.6%

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

      10. metadata-eval 78.6%

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

   6. Simplified 78.6%

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

   if 1.9e151 < l

   1. Initial program 11.0%

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

   3. Simplified 28.4%

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

      1. pow1 28.4%

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

   5. Applied egg-rr 28.4%

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

      1. unpow1 28.4%

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

      2. associate-*l* 30.2%

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

      3. *-commutative 30.2%

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

      4. fma-udef 30.2%

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

      5. associate-*l* 30.2%

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

      6. fma-def 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in U* around inf 10.2%

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

      1. *-commutative 10.2%

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

      2. unpow2 10.2%

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

      3. times-frac 11.0%

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

      4. unpow2 11.0%

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

      5. associate-/l* 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in l around inf 60.9%

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

       1. sub-neg 60.9%

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

       2. unpow2 60.9%

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

       3. times-frac 68.0%

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

       4. associate-*r/ 68.0%

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

       5. metadata-eval 68.0%

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

       6. distribute-neg-frac 68.0%

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

       7. metadata-eval 68.0%

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

   13. Simplified 68.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right) + \frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot \frac{\frac{U*}{Om}}{Om} - \frac{n}{Om} \cdot \frac{U}{Om}\right) - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 6: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-2, U \cdot \left(\ell \cdot \frac{\ell \cdot n}{Om}\right), n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.8e-77)
   (sqrt (* 2.0 (fma -2.0 (* U (* l (/ (* l n) Om))) (* n (* U t)))))
   (if (<= l 2.7e+151)
     (sqrt
      (*
       2.0
       (*
        U
        (fma
         n
         t
         (* (* l l) (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt (* U (* n (+ (/ -2.0 Om) (* (/ n Om) (/ U* Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.8e-77) {
		tmp = sqrt((2.0 * fma(-2.0, (U * (l * ((l * n) / Om))), (n * (U * t)))));
	} else if (l <= 2.7e+151) {
		tmp = sqrt((2.0 * (U * fma(n, t, ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.8e-77)
		tmp = sqrt(Float64(2.0 * fma(-2.0, Float64(U * Float64(l * Float64(Float64(l * n) / Om))), Float64(n * Float64(U * t)))));
	elseif (l <= 2.7e+151)
		tmp = sqrt(Float64(2.0 * Float64(U * fma(n, t, Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.8e-77], N[Sqrt[N[(2.0 * N[(-2.0 * N[(U * N[(l * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.7e+151], N[Sqrt[N[(2.0 * N[(U * N[(n * t + N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.79999999999999966e-77

   1. Initial program 54.0%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in l around 0 42.6%

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

      1. distribute-lft-out 42.6%

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

      2. fma-def 43.8%

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

      3. unpow2 43.8%

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

      4. sub-neg 43.8%

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

      5. unpow2 43.8%

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

      6. times-frac 51.1%

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

      7. associate-*r/ 51.1%

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

      8. metadata-eval 51.1%

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

      9. distribute-neg-frac 51.1%

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

      10. metadata-eval 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in Om around inf 45.0%

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

      1. fma-def 45.0%

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

      2. associate-*r/ 45.9%

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

      3. associate-*r/ 46.8%

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

      4. unpow2 46.8%

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

      5. associate-*l* 49.1%

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

      6. associate-*r/ 50.2%

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

      7. associate-*r* 49.2%

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

      8. *-commutative 49.2%

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

      9. associate-*r* 52.6%

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

   9. Simplified 52.6%

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

   if 6.79999999999999966e-77 < l < 2.7000000000000001e151

   1. Initial program 66.1%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in l around 0 70.5%

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

      1. distribute-lft-out 70.5%

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

      2. fma-def 70.5%

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

      3. unpow2 70.5%

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

      4. sub-neg 70.5%

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

      5. unpow2 70.5%

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

      6. times-frac 72.9%

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

      7. associate-*r/ 72.9%

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

      8. metadata-eval 72.9%

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

      9. distribute-neg-frac 72.9%

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

      10. metadata-eval 72.9%

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

   6. Simplified 72.9%

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

   if 2.7000000000000001e151 < l

   1. Initial program 11.0%

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

   3. Simplified 28.4%

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

      1. pow1 28.4%

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

   5. Applied egg-rr 28.4%

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

      1. unpow1 28.4%

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

      2. associate-*l* 30.2%

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

      3. *-commutative 30.2%

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

      4. fma-udef 30.2%

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

      5. associate-*l* 30.2%

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

      6. fma-def 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in U* around inf 10.2%

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

      1. *-commutative 10.2%

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

      2. unpow2 10.2%

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

      3. times-frac 11.0%

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

      4. unpow2 11.0%

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

      5. associate-/l* 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in l around inf 60.9%

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

       1. sub-neg 60.9%

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

       2. unpow2 60.9%

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

       3. times-frac 68.0%

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

       4. associate-*r/ 68.0%

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

       5. metadata-eval 68.0%

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

       6. distribute-neg-frac 68.0%

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

       7. metadata-eval 68.0%

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

   13. Simplified 68.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-2, U \cdot \left(\ell \cdot \frac{\ell \cdot n}{Om}\right), n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 7: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7e-76)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ (+ t (* -2.0 (/ (* l l) Om))) (* n (* (pow (/ l Om) 2.0) (- U* U))))))
   (if (<= l 2.8e+151)
     (sqrt
      (*
       2.0
       (*
        U
        (fma
         n
         t
         (* (* l l) (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt (* U (* n (+ (/ -2.0 Om) (* (/ n Om) (/ U* Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7e-76) {
		tmp = sqrt(((2.0 * (n * U)) * ((t + (-2.0 * ((l * l) / Om))) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	} else if (l <= 2.8e+151) {
		tmp = sqrt((2.0 * (U * fma(n, t, ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7e-76)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))));
	elseif (l <= 2.8e+151)
		tmp = sqrt(Float64(2.0 * Float64(U * fma(n, t, Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7e-76], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $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] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.8e+151], N[Sqrt[N[(2.0 * N[(U * N[(n * t + N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.99999999999999995e-76

   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 54.5%

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

   if 6.99999999999999995e-76 < l < 2.79999999999999987e151

   1. Initial program 66.1%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in l around 0 70.5%

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

      1. distribute-lft-out 70.5%

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

      2. fma-def 70.5%

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

      3. unpow2 70.5%

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

      4. sub-neg 70.5%

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

      5. unpow2 70.5%

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

      6. times-frac 72.9%

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

      7. associate-*r/ 72.9%

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

      8. metadata-eval 72.9%

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

      9. distribute-neg-frac 72.9%

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

      10. metadata-eval 72.9%

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

   6. Simplified 72.9%

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

   if 2.79999999999999987e151 < l

   1. Initial program 11.0%

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

   3. Simplified 28.4%

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

      1. pow1 28.4%

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

   5. Applied egg-rr 28.4%

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

      1. unpow1 28.4%

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

      2. associate-*l* 30.2%

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

      3. *-commutative 30.2%

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

      4. fma-udef 30.2%

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

      5. associate-*l* 30.2%

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

      6. fma-def 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in U* around inf 10.2%

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

      1. *-commutative 10.2%

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

      2. unpow2 10.2%

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

      3. times-frac 11.0%

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

      4. unpow2 11.0%

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

      5. associate-/l* 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in l around inf 60.9%

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

       1. sub-neg 60.9%

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

       2. unpow2 60.9%

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

       3. times-frac 68.0%

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

       4. associate-*r/ 68.0%

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

       5. metadata-eval 68.0%

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

       6. distribute-neg-frac 68.0%

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

       7. metadata-eval 68.0%

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

   13. Simplified 68.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \mathsf{fma}\left(n, t, \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 8: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e-21)
   (sqrt (* 2.0 (* n (* U (+ t (* -2.0 (/ l (/ Om l))))))))
   (if (<= l 1.45e+144)
     (sqrt
      (*
       2.0
       (*
        (* n U)
        (fma (* l l) (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)) t))))
     (*
      (* l (sqrt 2.0))
      (sqrt (* U (* n (+ (/ -2.0 Om) (* (/ n Om) (/ U* Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e-21) {
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	} else if (l <= 1.45e+144) {
		tmp = sqrt((2.0 * ((n * U) * fma((l * l), (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)), t))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e-21)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))))));
	elseif (l <= 1.45e+144)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(Float64(l * l), Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om)), t))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e-21], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.45e+144], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.15e-21

   1. Initial program 54.2%

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

   3. Simplified 57.2%

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

      1. pow1 57.2%

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

   5. Applied egg-rr 57.2%

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

      1. unpow1 57.2%

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

      2. associate-*l* 59.7%

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

      3. *-commutative 59.7%

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

      4. fma-udef 59.7%

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

      5. associate-*l* 59.7%

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

      6. fma-def 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in n around 0 49.5%

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

      1. unpow2 49.5%

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

      2. associate-/l* 51.5%

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

   10. Simplified 51.5%

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

   if 1.15e-21 < l < 1.44999999999999999e144

   1. Initial program 67.7%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in l around 0 71.3%

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

      1. +-commutative 71.3%

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

      2. fma-def 71.3%

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

      3. unpow2 71.3%

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

      4. sub-neg 71.3%

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

      5. unpow2 71.3%

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

      6. times-frac 74.6%

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

      7. associate-*r/ 74.6%

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

      8. metadata-eval 74.6%

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

      9. distribute-neg-frac 74.6%

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

      10. metadata-eval 74.6%

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

   6. Simplified 74.6%

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

   if 1.44999999999999999e144 < l

   1. Initial program 13.8%

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

   3. Simplified 30.6%

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

      1. pow1 30.6%

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

   5. Applied egg-rr 30.6%

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

      1. unpow1 30.6%

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

      2. associate-*l* 32.4%

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

      3. *-commutative 32.4%

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

      4. fma-udef 32.4%

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

      5. associate-*l* 32.4%

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

      6. fma-def 32.4%

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

   7. Simplified 32.4%

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

   8. Taylor expanded in U* around inf 13.0%

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

      1. *-commutative 13.0%

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

      2. unpow2 13.0%

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

      3. times-frac 13.8%

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

      4. unpow2 13.8%

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

      5. associate-/l* 32.5%

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

   10. Simplified 32.5%

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

   11. Taylor expanded in l around inf 62.1%

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

       1. sub-neg 62.1%

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

       2. unpow2 62.1%

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

       3. times-frac 69.0%

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

       4. associate-*r/ 69.0%

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

       5. metadata-eval 69.0%

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

       6. distribute-neg-frac 69.0%

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

       7. metadata-eval 69.0%

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

   13. Simplified 69.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 9: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{U*}{Om \cdot \frac{Om}{n}} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.3e-183)
   (* (sqrt 2.0) (sqrt (* t (* n U))))
   (if (<= l 2.05e+138)
     (sqrt
      (*
       2.0
       (* n (* U (+ t (* (* l l) (- (/ U* (* Om (/ Om n))) (/ 2.0 Om))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.3e-183) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else if (l <= 2.05e+138) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.3d-183) then
        tmp = sqrt(2.0d0) * sqrt((t * (n * u)))
    else if (l <= 2.05d+138) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * l) * ((u_42 / (om * (om / n))) - (2.0d0 / om))))))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((u * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.3e-183) {
		tmp = Math.sqrt(2.0) * Math.sqrt((t * (n * U)));
	} else if (l <= 2.05e+138) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.3e-183:
		tmp = math.sqrt(2.0) * math.sqrt((t * (n * U)))
	elif l <= 2.05e+138:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.3e-183)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	elseif (l <= 2.05e+138)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(U_42_ / Float64(Om * Float64(Om / n))) - Float64(2.0 / Om))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.3e-183)
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	elseif (l <= 2.05e+138)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.3e-183], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e+138], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(U$42$ / N[(Om * N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.2999999999999999e-183

   1. Initial program 53.0%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-*r* 39.3%

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

      3. *-commutative 39.3%

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

   6. Simplified 39.3%

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

   if 1.2999999999999999e-183 < l < 2.0499999999999999e138

   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 65.0%

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

      1. pow1 65.0%

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

   5. Applied egg-rr 65.0%

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

      1. unpow1 65.0%

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

      2. associate-*l* 63.5%

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

      3. *-commutative 63.5%

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

      4. fma-udef 63.5%

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

      5. associate-*l* 63.5%

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

      6. fma-def 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in U* around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. unpow2 56.4%

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

      3. times-frac 63.5%

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

      4. unpow2 63.5%

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

      5. associate-/l* 63.5%

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

   10. Simplified 63.5%

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

   11. Taylor expanded in l around 0 60.1%

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

       1. unpow2 60.1%

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

       2. associate-/l* 60.3%

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

       3. unpow2 60.3%

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

       4. associate-*r/ 60.4%

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

       5. associate-*r/ 60.4%

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

       6. metadata-eval 60.4%

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

   13. Simplified 60.4%

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

   if 2.0499999999999999e138 < l

   1. Initial program 18.0%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in l around inf 58.5%

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

      1. sub-neg 58.5%

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

      2. unpow2 58.5%

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

      3. times-frac 64.7%

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

      4. associate-*r/ 64.7%

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

      5. metadata-eval 64.7%

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

      6. distribute-neg-frac 64.7%

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

      7. metadata-eval 64.7%

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

   6. Simplified 64.7%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{U*}{Om \cdot \frac{Om}{n}} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 10: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.4e-162)
   (* (sqrt 2.0) (sqrt (* t (* n U))))
   (if (<= l 6.2e+140)
     (sqrt
      (*
       2.0
       (* n (* U (- t (* (* l l) (- (/ 2.0 Om) (/ n (/ Om (/ U* Om))))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt (* U (* n (+ (/ -2.0 Om) (* (/ n Om) (/ U* Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.4e-162) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else if (l <= 6.2e+140) {
		tmp = sqrt((2.0 * (n * (U * (t - ((l * l) * ((2.0 / Om) - (n / (Om / (U_42_ / Om))))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.4d-162) then
        tmp = sqrt(2.0d0) * sqrt((t * (n * u)))
    else if (l <= 6.2d+140) then
        tmp = sqrt((2.0d0 * (n * (u * (t - ((l * l) * ((2.0d0 / om) - (n / (om / (u_42 / om))))))))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((u * (n * (((-2.0d0) / om) + ((n / om) * (u_42 / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.4e-162) {
		tmp = Math.sqrt(2.0) * Math.sqrt((t * (n * U)));
	} else if (l <= 6.2e+140) {
		tmp = Math.sqrt((2.0 * (n * (U * (t - ((l * l) * ((2.0 / Om) - (n / (Om / (U_42_ / Om))))))))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.4e-162:
		tmp = math.sqrt(2.0) * math.sqrt((t * (n * U)))
	elif l <= 6.2e+140:
		tmp = math.sqrt((2.0 * (n * (U * (t - ((l * l) * ((2.0 / Om) - (n / (Om / (U_42_ / Om))))))))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.4e-162)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	elseif (l <= 6.2e+140)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t - Float64(Float64(l * l) * Float64(Float64(2.0 / Om) - Float64(n / Float64(Om / Float64(U_42_ / Om))))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(-2.0 / Om) + Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.4e-162)
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	elseif (l <= 6.2e+140)
		tmp = sqrt((2.0 * (n * (U * (t - ((l * l) * ((2.0 / Om) - (n / (Om / (U_42_ / Om))))))))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((U * (n * ((-2.0 / Om) + ((n / Om) * (U_42_ / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.4e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+140], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t - N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - N[(n / N[(Om / N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(-2.0 / Om), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.3999999999999998e-162

   1. Initial program 53.6%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in l around 0 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r* 40.0%

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

      3. *-commutative 40.0%

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

   6. Simplified 40.0%

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

   if 4.3999999999999998e-162 < l < 6.2000000000000001e140

   1. Initial program 64.4%

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

   3. Simplified 64.4%

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

      1. pow1 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. unpow1 64.4%

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

      2. associate-*l* 62.8%

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

      3. *-commutative 62.8%

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

      4. fma-udef 62.8%

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

      5. associate-*l* 62.8%

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

      6. fma-def 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in l around 0 57.6%

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

      1. unpow2 57.6%

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

      2. associate-/l* 59.5%

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

      3. unpow2 59.5%

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

      4. associate-*r/ 59.5%

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

      5. metadata-eval 59.5%

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

   10. Simplified 59.5%

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

   11. Taylor expanded in U* around inf 59.6%

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

       1. unpow2 59.6%

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

       2. associate-/l* 61.4%

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

   13. Simplified 61.4%

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

   if 6.2000000000000001e140 < l

   1. Initial program 15.8%

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

   3. Simplified 28.3%

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

      1. pow1 28.3%

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

   5. Applied egg-rr 28.3%

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

      1. unpow1 28.3%

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

      2. associate-*l* 29.9%

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

      3. *-commutative 29.9%

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

      4. fma-udef 29.9%

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

      5. associate-*l* 29.9%

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

      6. fma-def 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in U* around inf 12.0%

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

      1. *-commutative 12.0%

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

      2. unpow2 12.0%

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

      3. times-frac 13.0%

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

      4. unpow2 13.0%

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

      5. associate-/l* 30.1%

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

   10. Simplified 30.1%

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

   11. Taylor expanded in l around inf 60.1%

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

       1. sub-neg 60.1%

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

       2. unpow2 60.1%

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

       3. times-frac 66.4%

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

       4. associate-*r/ 66.4%

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

       5. metadata-eval 66.4%

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

       6. distribute-neg-frac 66.4%

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

       7. metadata-eval 66.4%

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

   13. Simplified 66.4%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{-2}{Om} + \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 11: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-2, U \cdot \left(\ell \cdot \frac{\ell \cdot n}{Om}\right), n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.2e-162)
   (* (sqrt 2.0) (sqrt (* t (* n U))))
   (if (<= l 1.85e+143)
     (sqrt
      (*
       2.0
       (* n (* U (- t (* (* l l) (- (/ 2.0 Om) (/ n (/ Om (/ U* Om))))))))))
     (sqrt (* 2.0 (fma -2.0 (* U (* l (/ (* l n) Om))) (* n (* U t))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.2e-162) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else if (l <= 1.85e+143) {
		tmp = sqrt((2.0 * (n * (U * (t - ((l * l) * ((2.0 / Om) - (n / (Om / (U_42_ / Om))))))))));
	} else {
		tmp = sqrt((2.0 * fma(-2.0, (U * (l * ((l * n) / Om))), (n * (U * t)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.2e-162)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	elseif (l <= 1.85e+143)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t - Float64(Float64(l * l) * Float64(Float64(2.0 / Om) - Float64(n / Float64(Om / Float64(U_42_ / Om))))))))));
	else
		tmp = sqrt(Float64(2.0 * fma(-2.0, Float64(U * Float64(l * Float64(Float64(l * n) / Om))), Float64(n * Float64(U * t)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.2e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.85e+143], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t - N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - N[(n / N[(Om / N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(-2.0 * N[(U * N[(l * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.1999999999999999e-162

   1. Initial program 53.6%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in l around 0 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r* 40.0%

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

      3. *-commutative 40.0%

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

   6. Simplified 40.0%

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

   if 5.1999999999999999e-162 < l < 1.8500000000000001e143

   1. Initial program 64.0%

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

   3. Simplified 62.3%

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

      1. pow1 62.3%

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

   5. Applied egg-rr 62.3%

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

      1. unpow1 62.3%

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

      2. associate-*l* 60.7%

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

      3. *-commutative 60.7%

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

      4. fma-udef 60.7%

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

      5. associate-*l* 60.7%

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

      6. fma-def 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in l around 0 55.8%

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

      1. unpow2 55.8%

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

      2. associate-/l* 57.6%

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

      3. unpow2 57.6%

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

      4. associate-*r/ 57.6%

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

      5. metadata-eval 57.6%

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

   10. Simplified 57.6%

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

   11. Taylor expanded in U* around inf 57.7%

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

       1. unpow2 57.7%

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

       2. associate-/l* 59.5%

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

   13. Simplified 59.5%

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

   if 1.8500000000000001e143 < l

   1. Initial program 13.5%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in l around 0 29.3%

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

      1. distribute-lft-out 29.3%

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

      2. fma-def 29.3%

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

      3. unpow2 29.3%

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

      4. sub-neg 29.3%

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

      5. unpow2 29.3%

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

      6. times-frac 29.5%

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

      7. associate-*r/ 29.5%

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

      8. metadata-eval 29.5%

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

      9. distribute-neg-frac 29.5%

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

      10. metadata-eval 29.5%

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

   6. Simplified 29.5%

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

   7. Taylor expanded in Om around inf 20.8%

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

      1. fma-def 20.8%

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

      2. associate-*r/ 18.0%

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

      3. associate-*r/ 17.2%

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

      4. unpow2 17.2%

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

      5. associate-*l* 34.2%

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

      6. associate-*r/ 44.7%

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

      7. associate-*r* 44.1%

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

      8. *-commutative 44.1%

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

      9. associate-*r* 44.7%

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

   9. Simplified 44.7%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{\frac{Om}{\frac{U*}{Om}}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-2, U \cdot \left(\ell \cdot \frac{\ell \cdot n}{Om}\right), n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 12: 51.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{U*}{Om \cdot \frac{Om}{n}} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.75e-182)
   (* (sqrt 2.0) (sqrt (* t (* n U))))
   (if (<= l 1.2e+124)
     (sqrt
      (*
       2.0
       (* n (* U (+ t (* (* l l) (- (/ U* (* Om (/ Om n))) (/ 2.0 Om))))))))
     (sqrt
      (*
       2.0
       (*
        U
        (* (* l l) (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.75e-182) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else if (l <= 1.2e+124) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.75d-182) then
        tmp = sqrt(2.0d0) * sqrt((t * (n * u)))
    else if (l <= 1.2d+124) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * l) * ((u_42 / (om * (om / n))) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((l * l) * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.75e-182) {
		tmp = Math.sqrt(2.0) * Math.sqrt((t * (n * U)));
	} else if (l <= 1.2e+124) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.75e-182:
		tmp = math.sqrt(2.0) * math.sqrt((t * (n * U)))
	elif l <= 1.2e+124:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.75e-182)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	elseif (l <= 1.2e+124)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(U_42_ / Float64(Om * Float64(Om / n))) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.75e-182)
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	elseif (l <= 1.2e+124)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.75e-182], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+124], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(U$42$ / N[(Om * N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.74999999999999992e-182

   1. Initial program 53.0%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in l around 0 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-*r* 39.3%

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

      3. *-commutative 39.3%

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

   6. Simplified 39.3%

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

   if 1.74999999999999992e-182 < l < 1.20000000000000003e124

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

      1. pow1 66.8%

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

   5. Applied egg-rr 66.8%

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

      1. unpow1 66.8%

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

      2. associate-*l* 65.2%

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

      3. *-commutative 65.2%

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

      4. fma-udef 65.2%

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

      5. associate-*l* 65.2%

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

      6. fma-def 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in U* around inf 57.7%

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

      1. *-commutative 57.7%

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

      2. unpow2 57.7%

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

      3. times-frac 65.2%

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

      4. unpow2 65.2%

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

      5. associate-/l* 65.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in l around 0 61.4%

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

       1. unpow2 61.4%

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

       2. associate-/l* 61.6%

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

       3. unpow2 61.6%

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

       4. associate-*r/ 61.7%

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

       5. associate-*r/ 61.7%

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

       6. metadata-eval 61.7%

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

   13. Simplified 61.7%

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

   if 1.20000000000000003e124 < l

   1. Initial program 19.2%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in l around inf 30.6%

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

      1. unpow2 30.6%

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

      2. sub-neg 30.6%

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

      3. unpow2 30.6%

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

      4. times-frac 33.2%

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

      5. associate-*r/ 33.2%

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

      6. metadata-eval 33.2%

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

      7. distribute-neg-frac 33.2%

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

      8. metadata-eval 33.2%

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

   6. Simplified 33.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{U*}{Om \cdot \frac{Om}{n}} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13: 51.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{U*}{Om \cdot \frac{Om}{n}} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.9e-159)
   (sqrt (* 2.0 (* n (* U t))))
   (if (<= l 4.4e+124)
     (sqrt
      (*
       2.0
       (* n (* U (+ t (* (* l l) (- (/ U* (* Om (/ Om n))) (/ 2.0 Om))))))))
     (sqrt
      (*
       2.0
       (*
        U
        (* (* l l) (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.9e-159) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else if (l <= 4.4e+124) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 3.9d-159) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else if (l <= 4.4d+124) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * l) * ((u_42 / (om * (om / n))) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((l * l) * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.9e-159) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else if (l <= 4.4e+124) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.9e-159:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	elif l <= 4.4e+124:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.9e-159)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	elseif (l <= 4.4e+124)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(U_42_ / Float64(Om * Float64(Om / n))) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.9e-159)
		tmp = sqrt((2.0 * (n * (U * t))));
	elseif (l <= 4.4e+124)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * ((U_42_ / (Om * (Om / n))) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.9e-159], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.4e+124], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(U$42$ / N[(Om * N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.89999999999999977e-159

   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 57.3%

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

      1. pow1 57.3%

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

   5. Applied egg-rr 57.3%

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

      1. unpow1 57.3%

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

      2. associate-*l* 60.2%

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

      3. *-commutative 60.2%

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

      4. fma-udef 60.2%

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

      5. associate-*l* 60.2%

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

      6. fma-def 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in l around 0 42.4%

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

   if 3.89999999999999977e-159 < l < 4.4000000000000002e124

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

      1. pow1 64.9%

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

   5. Applied egg-rr 64.9%

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

      1. unpow1 64.9%

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

      2. associate-*l* 63.1%

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

      3. *-commutative 63.1%

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

      4. fma-udef 63.1%

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

      5. associate-*l* 63.1%

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

      6. fma-def 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in U* around inf 55.2%

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

      1. *-commutative 55.2%

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

      2. unpow2 55.2%

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

      3. times-frac 63.2%

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

      4. unpow2 63.2%

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

      5. associate-/l* 63.2%

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

   10. Simplified 63.2%

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

   11. Taylor expanded in l around 0 59.1%

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

       1. unpow2 59.1%

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

       2. associate-/l* 61.3%

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

       3. unpow2 61.3%

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

       4. associate-*r/ 61.3%

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

       5. associate-*r/ 61.3%

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

       6. metadata-eval 61.3%

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

   13. Simplified 61.3%

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

   if 4.4000000000000002e124 < l

   1. Initial program 19.2%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in l around inf 30.6%

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

      1. unpow2 30.6%

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

      2. sub-neg 30.6%

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

      3. unpow2 30.6%

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

      4. times-frac 33.2%

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

      5. associate-*r/ 33.2%

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

      6. metadata-eval 33.2%

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

      7. distribute-neg-frac 33.2%

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

      8. metadata-eval 33.2%

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

   6. Simplified 33.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{U*}{Om \cdot \frac{Om}{n}} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14: 48.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{U*}{\frac{Om \cdot Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 6.8e+84)
   (sqrt (* 2.0 (* n (* U (+ t (* -2.0 (/ l (/ Om l))))))))
   (sqrt (* 2.0 (* n (* U (+ t (/ U* (/ (* Om Om) (* n (* l l)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 6.8e+84) {
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	} else {
		tmp = sqrt((2.0 * (n * (U * (t + (U_42_ / ((Om * Om) / (n * (l * l)))))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 6.8d+84) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((-2.0d0) * (l / (om / l))))))))
    else
        tmp = sqrt((2.0d0 * (n * (u * (t + (u_42 / ((om * om) / (n * (l * l)))))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 6.8e+84) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (U_42_ / ((Om * Om) / (n * (l * l)))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 6.8e+84:
		tmp = math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * (t + (U_42_ / ((Om * Om) / (n * (l * l)))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 6.8e+84)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(U_42_ / Float64(Float64(Om * Om) / Float64(n * Float64(l * l)))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 6.8e+84)
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
	else
		tmp = sqrt((2.0 * (n * (U * (t + (U_42_ / ((Om * Om) / (n * (l * l)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 6.8e+84], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(U$42$ / N[(N[(Om * Om), $MachinePrecision] / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.7999999999999996e84

   1. Initial program 50.8%

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

   3. Simplified 54.1%

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

      1. pow1 54.1%

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

   5. Applied egg-rr 54.1%

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

      1. unpow1 54.1%

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

      2. associate-*l* 55.9%

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

      3. *-commutative 55.9%

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

      4. fma-udef 55.9%

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

      5. associate-*l* 55.9%

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

      6. fma-def 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in n around 0 47.2%

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

      1. unpow2 47.2%

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

      2. associate-/l* 51.2%

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

   10. Simplified 51.2%

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

   if 6.7999999999999996e84 < n

   1. Initial program 50.4%

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

   3. Simplified 57.9%

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

      1. pow1 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. unpow1 57.9%

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

      2. associate-*l* 59.5%

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

      3. *-commutative 59.5%

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

      4. fma-udef 59.5%

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

      5. associate-*l* 59.5%

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

      6. fma-def 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in l around 0 38.0%

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

      1. unpow2 38.0%

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

      2. associate-/l* 49.8%

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

      3. unpow2 49.8%

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

      4. associate-*r/ 49.8%

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

      5. metadata-eval 49.8%

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

   10. Simplified 49.8%

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

   11. Taylor expanded in U* around inf 49.9%

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

       1. associate-/l* 49.9%

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

       2. unpow2 49.9%

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

       3. *-commutative 49.9%

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

       4. unpow2 49.9%

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

   13. Simplified 49.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{U*}{\frac{Om \cdot Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right)}\\ \end{array} \]

Alternative 15: 38.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7e+113)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* -2.0 (/ U (/ Om (* n (* l l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7e+113) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 7d+113) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * ((-2.0d0) * (u / (om / (n * (l * l)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7e+113) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7e+113:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7e+113)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(-2.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 7e+113)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7e+113], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(-2.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.0000000000000001e113

   1. Initial program 56.2%

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

   3. Simplified 58.8%

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

      1. pow1 58.8%

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

   5. Applied egg-rr 58.8%

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

      1. unpow1 58.8%

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

      2. associate-*l* 60.7%

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

      3. *-commutative 60.7%

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

      4. fma-udef 60.7%

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

      5. associate-*l* 60.7%

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

      6. fma-def 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in l around 0 42.8%

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

   if 7.0000000000000001e113 < l

   1. Initial program 21.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in l around inf 29.9%

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

      1. unpow2 29.9%

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

      2. sub-neg 29.9%

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

      3. unpow2 29.9%

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

      4. times-frac 32.5%

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

      5. associate-*r/ 32.5%

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

      6. metadata-eval 32.5%

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

      7. distribute-neg-frac 32.5%

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

      8. metadata-eval 32.5%

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

   6. Simplified 32.5%

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

   7. Taylor expanded in n around 0 20.1%

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

      1. associate-/l* 17.9%

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

      2. *-commutative 17.9%

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

      3. unpow2 17.9%

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

   9. Simplified 17.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 16: 39.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{n}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.55e+23)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* U (* -2.0 (/ (* l l) (/ Om n))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.55e+23) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (U * (-2.0 * ((l * l) / (Om / n))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.55d+23) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (u * ((-2.0d0) * ((l * l) / (om / n))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.55e+23) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (-2.0 * ((l * l) / (Om / n))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.55e+23:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (U * (-2.0 * ((l * l) / (Om / n))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.55e+23)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(-2.0 * Float64(Float64(l * l) / Float64(Om / n))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.55e+23)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (U * (-2.0 * ((l * l) / (Om / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.55e+23], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(-2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.54999999999999985e23

   1. Initial program 55.0%

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

   3. Simplified 57.8%

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

      1. pow1 57.8%

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

   5. Applied egg-rr 57.8%

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

      1. unpow1 57.8%

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

      2. associate-*l* 59.7%

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

      3. *-commutative 59.7%

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

      4. fma-udef 59.7%

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

      5. associate-*l* 59.7%

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

      6. fma-def 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in l around 0 42.5%

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

   if 1.54999999999999985e23 < l

   1. Initial program 33.0%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in l around inf 32.5%

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

      1. unpow2 32.5%

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

      2. sub-neg 32.5%

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

      3. unpow2 32.5%

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

      4. times-frac 34.6%

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

      5. associate-*r/ 34.6%

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

      6. metadata-eval 34.6%

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

      7. distribute-neg-frac 34.6%

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

      8. metadata-eval 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in n around 0 17.0%

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

      1. associate-/l* 20.1%

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

      2. unpow2 20.1%

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

   9. Simplified 20.1%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{n}}\right)\right)}\\ \end{array} \]

Alternative 17: 39.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{-2}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.05e+23)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* U (* (* l l) (* n (/ -2.0 Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.05e+23) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (-2.0 / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.05d+23) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (u * ((l * l) * (n * ((-2.0d0) / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.05e+23) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((l * l) * (n * (-2.0 / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.05e+23:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (U * ((l * l) * (n * (-2.0 / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.05e+23)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(-2.0 / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.05e+23)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (-2.0 / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.05e+23], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0500000000000001e23

   1. Initial program 55.0%

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

   3. Simplified 57.8%

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

      1. pow1 57.8%

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

   5. Applied egg-rr 57.8%

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

      1. unpow1 57.8%

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

      2. associate-*l* 59.7%

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

      3. *-commutative 59.7%

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

      4. fma-udef 59.7%

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

      5. associate-*l* 59.7%

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

      6. fma-def 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in l around 0 42.5%

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

   if 1.0500000000000001e23 < l

   1. Initial program 33.0%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in l around inf 32.5%

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

      1. unpow2 32.5%

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

      2. sub-neg 32.5%

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

      3. unpow2 32.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)} \]

      4. times-frac 34.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)} \]

      5. associate-*r/ 34.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)\right)} \]

      6. metadata-eval 34.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)\right)} \]

      7. distribute-neg-frac 34.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \color{blue}{\frac{-2}{Om}}\right)\right)\right)\right)} \]

      8. metadata-eval 34.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

   6. Simplified 34.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}} \]

   7. Taylor expanded in n around 0 20.2%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(-2 \cdot \frac{n}{Om}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 20.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-2 \cdot n}{Om}}\right)\right)} \]

      2. associate-*l/ 20.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{-2}{Om} \cdot n\right)}\right)\right)} \]

      3. *-commutative 20.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot \frac{-2}{Om}\right)}\right)\right)} \]

   9. Simplified 20.3%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot \frac{-2}{Om}\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{-2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 18: 47.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* 2.0 (* n (* U (+ t (* -2.0 (/ l (/ Om l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (n * (u * (t + ((-2.0d0) * (l / (om / l))))))))
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))))

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))))))
end

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l / (Om / l))))))));
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 54.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
4. Step-by-step derivation

   1. pow1 54.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

5. Applied egg-rr 54.7%

   \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
6. Step-by-step derivation

   1. unpow1 54.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   2. associate-*l* 56.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

   3. *-commutative 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \color{blue}{\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}}, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

   4. fma-udef 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t}\right)\right)\right)} \]

   5. associate-*l* 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} + t\right)\right)\right)} \]

   6. fma-def 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)}\right)\right)\right)} \]

7. Simplified 56.5%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}} \]

8. Taylor expanded in n around 0 45.5%

   \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]
9. Step-by-step derivation

   1. unpow2 45.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]

   2. associate-/l* 49.2%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right)\right)\right)} \]

10. Simplified 49.2%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\right)\right)} \]

11. Final simplification 49.2%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)} \]

Alternative 19: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e-21)
   (sqrt (* 2.0 (* n (* U t))))
   (pow (* 2.0 (* t (* n U))) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-21) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2d-21) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-21) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2e-21:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2e-21)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2e-21)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2e-21], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.99999999999999982e-21

   1. Initial program 54.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 57.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 57.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 57.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 59.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. *-commutative 59.7%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \color{blue}{\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}}, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      4. fma-udef 59.7%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t}\right)\right)\right)} \]

      5. associate-*l* 59.7%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} + t\right)\right)\right)} \]

      6. fma-def 59.7%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)}\right)\right)\right)} \]

   7. Simplified 59.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in l around 0 42.4%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{t}\right)\right)} \]

   if 1.99999999999999982e-21 < l

   1. Initial program 39.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 46.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around 0 21.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 21.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 22.8%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{0.5} \]

      3. *-commutative 22.8%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)\right)}^{0.5} \]

      4. *-commutative 22.8%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(t \cdot \left(n \cdot U\right)\right)}\right)}^{0.5} \]

   6. Applied egg-rr 22.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 20: 35.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq -8.2 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -8.2e+164)
   (sqrt (* 2.0 (* t (* n U))))
   (sqrt (* 2.0 (* n (* U t))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -8.2e+164) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-8.2d+164)) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt((2.0d0 * (n * (u * t))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -8.2e+164) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -8.2e+164:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -8.2e+164)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -8.2e+164)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt((2.0 * (n * (U * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -8.2e+164], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -8.20000000000000032e164

   1. Initial program 84.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around 0 64.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 68.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 68.9%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   6. Simplified 68.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

   if -8.20000000000000032e164 < U

   1. Initial program 48.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 52.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 52.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 52.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 56.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. *-commutative 56.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \color{blue}{\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}}, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      4. fma-udef 56.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t}\right)\right)\right)} \]

      5. associate-*l* 56.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} + t\right)\right)\right)} \]

      6. fma-def 56.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)}\right)\right)\right)} \]

   7. Simplified 56.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in l around 0 36.3%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -8.2 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 21: 36.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 54.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

4. Taylor expanded in l around 0 35.2%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

5. Final simplification 35.2%

   \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Alternative 22: 35.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* n (* U t)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (n * (U * t))));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (n * (u * t))))
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (n * (U * t))));
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (n * (U * t))))

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(n * Float64(U * t))))
end

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (n * (U * t))));
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 54.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
4. Step-by-step derivation

   1. pow1 54.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

5. Applied egg-rr 54.7%

   \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
6. Step-by-step derivation

   1. unpow1 54.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   2. associate-*l* 56.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

   3. *-commutative 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \color{blue}{\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}}, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

   4. fma-udef 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t}\right)\right)\right)} \]

   5. associate-*l* 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} + t\right)\right)\right)} \]

   6. fma-def 56.5%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \color{blue}{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)}\right)\right)\right)} \]

7. Simplified 56.5%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}} \]

8. Taylor expanded in l around 0 36.9%

   \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{t}\right)\right)} \]

9. Final simplification 36.9%

   \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)} \]

Reproduce

herbie shell --seed 2023292 
(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*))))))