Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 63.7%

Time: 21.2s

Alternatives: 18

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.3% 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: 63.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.55 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(U* - U, \frac{l\_m}{Om} \cdot n, l\_m \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.55e+165)
   (sqrt
    (*
     U
     (*
      (fma (/ l_m Om) (fma (- U* U) (* (/ l_m Om) n) (* l_m -2.0)) t)
      (* n 2.0))))
   (*
    (sqrt (/ (* (* U n) (fma n (/ (- U* U) Om) -2.0)) Om))
    (* l_m (sqrt 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.55e+165) {
		tmp = sqrt((U * (fma((l_m / Om), fma((U_42_ - U), ((l_m / Om) * n), (l_m * -2.0)), t) * (n * 2.0))));
	} else {
		tmp = sqrt((((U * n) * fma(n, ((U_42_ - U) / Om), -2.0)) / Om)) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.55e+165)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), fma(Float64(U_42_ - U), Float64(Float64(l_m / Om) * n), Float64(l_m * -2.0)), t) * Float64(n * 2.0))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(U * n) * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)) / Om)) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.55e+165], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.54999999999999988e165

   1. Initial program 47.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 51.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 59.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   if 3.54999999999999988e165 < l

   1. Initial program 28.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 33.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 47.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 61.6%

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

   10. Taylor expanded in l around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sqrt-lowering-sqrt.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. /-lowering-/.f64 N/A

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

       12. --lowering--.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. sqrt-lowering-sqrt.f64 73.6

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

   12. Simplified 73.6%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.55 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* - U, \frac{\ell}{Om} \cdot n, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(l\_m \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;l\_m \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;l\_m \leq 4.8 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot 2\right) \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om} \cdot \frac{\mathsf{fma}\left(U* - U, n, Om \cdot -2\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.5e-62)
   (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (/ (* U* (* l_m n)) Om) t))))
   (if (<= l_m 4.8e+76)
     (sqrt
      (*
       (* U (* n 2.0))
       (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
     (if (<= l_m 4.8e+144)
       (sqrt
        (*
         (/ (* (* U 2.0) (* n (* l_m l_m))) Om)
         (/ (fma (- U* U) n (* Om -2.0)) Om)))
       (*
        (sqrt (/ (* (* U n) (fma n (/ (- U* U) Om) -2.0)) Om))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.5e-62) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), ((U_42_ * (l_m * n)) / Om), t))));
	} else if (l_m <= 4.8e+76) {
		tmp = sqrt(((U * (n * 2.0)) * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else if (l_m <= 4.8e+144) {
		tmp = sqrt(((((U * 2.0) * (n * (l_m * l_m))) / Om) * (fma((U_42_ - U), n, (Om * -2.0)) / Om)));
	} else {
		tmp = sqrt((((U * n) * fma(n, ((U_42_ - U) / Om), -2.0)) / Om)) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4.5e-62)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(l_m * n)) / Om), t))));
	elseif (l_m <= 4.8e+76)
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	elseif (l_m <= 4.8e+144)
		tmp = sqrt(Float64(Float64(Float64(Float64(U * 2.0) * Float64(n * Float64(l_m * l_m))) / Om) * Float64(fma(Float64(U_42_ - U), n, Float64(Om * -2.0)) / Om)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(U * n) * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)) / Om)) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 4.5e-62], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 4.8e+76], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 4.8e+144], N[Sqrt[N[(N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(U$42$ - U), $MachinePrecision] * n + N[(Om * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 4.50000000000000018e-62

   1. Initial program 47.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 51.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 59.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U* around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 49.1

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

   9. Simplified 49.1%

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

   if 4.50000000000000018e-62 < l < 4.8e76

   1. Initial program 61.0%

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

   3. Taylor expanded in t around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. /-lowering-/.f64 N/A

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

   5. Simplified 75.5%

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

   if 4.8e76 < l < 4.8000000000000001e144

   1. Initial program 33.8%

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

   3. Taylor expanded in Om around 0

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

      1. /-lowering-/.f64 N/A

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

   5. Simplified 14.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(Om, \mathsf{fma}\left(Om, t, \left(\ell \cdot \ell\right) \cdot -2\right), \left(n \cdot \left(U - U*\right)\right) \cdot \left(0 - \ell \cdot \ell\right)\right)}{Om \cdot Om}}} \]

   6. Taylor expanded in l around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   8. Simplified 50.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 61.2%

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

   if 4.8000000000000001e144 < l

   1. Initial program 30.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 38.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 53.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 61.6%

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

   10. Taylor expanded in l around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sqrt-lowering-sqrt.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. /-lowering-/.f64 N/A

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

       12. --lowering--.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. sqrt-lowering-sqrt.f64 67.7

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

   12. Simplified 67.7%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot 2\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om} \cdot \frac{\mathsf{fma}\left(U* - U, n, Om \cdot -2\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(l\_m \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{if}\;n \leq -1.52 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right)\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* U (* (* n 2.0) (fma (/ l_m Om) (/ (* U* (* l_m n)) Om) t))))))
   (if (<= n -1.52e-62)
     t_1
     (if (<= n -4e-311)
       (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (* l_m -2.0) t))))
       (if (<= n 4.2e-54)
         (* (sqrt n) (sqrt (* (* U 2.0) (fma -2.0 (/ (* l_m l_m) Om) t))))
         t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U * ((n * 2.0) * fma((l_m / Om), ((U_42_ * (l_m * n)) / Om), t))));
	double tmp;
	if (n <= -1.52e-62) {
		tmp = t_1;
	} else if (n <= -4e-311) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), (l_m * -2.0), t))));
	} else if (n <= 4.2e-54) {
		tmp = sqrt(n) * sqrt(((U * 2.0) * fma(-2.0, ((l_m * l_m) / Om), t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(l_m * n)) / Om), t))))
	tmp = 0.0
	if (n <= -1.52e-62)
		tmp = t_1;
	elseif (n <= -4e-311)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(l_m * -2.0), t))));
	elseif (n <= 4.2e-54)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(U * 2.0) * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.52e-62], t$95$1, If[LessEqual[n, -4e-311], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.2e-54], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.52000000000000007e-62 or 4.2e-54 < n

   1. Initial program 55.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 61.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 66.8%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U* around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 60.4

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

   9. Simplified 60.4%

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

   if -1.52000000000000007e-62 < n < -3.99999999999979e-311

   1. Initial program 40.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 44.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 58.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 56.9

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

   9. Simplified 56.9%

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

   if -3.99999999999979e-311 < n < 4.2e-54

   1. Initial program 29.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 43.7%

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

   5. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 47.9

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

   7. Simplified 47.9%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.52 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(l\_m \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;l\_m \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(l\_m \cdot n\right) \cdot \left(U \cdot 2\right)\right) \cdot \left(l\_m \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8e-61)
   (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (/ (* U* (* l_m n)) Om) t))))
   (if (<= l_m 2.15e+77)
     (sqrt
      (*
       (* U (* n 2.0))
       (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
     (sqrt
      (/
       (* (* (* l_m n) (* U 2.0)) (* l_m (fma n (/ (- U* U) Om) -2.0)))
       Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8e-61) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), ((U_42_ * (l_m * n)) / Om), t))));
	} else if (l_m <= 2.15e+77) {
		tmp = sqrt(((U * (n * 2.0)) * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = sqrt(((((l_m * n) * (U * 2.0)) * (l_m * fma(n, ((U_42_ - U) / Om), -2.0))) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8e-61)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(l_m * n)) / Om), t))));
	elseif (l_m <= 2.15e+77)
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * n) * Float64(U * 2.0)) * Float64(l_m * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 8e-61], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.15e+77], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.0000000000000003e-61

   1. Initial program 47.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 51.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 59.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U* around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 49.1

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

   9. Simplified 49.1%

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

   if 8.0000000000000003e-61 < l < 2.14999999999999996e77

   1. Initial program 61.0%

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

   3. Taylor expanded in t around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. /-lowering-/.f64 N/A

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

   5. Simplified 75.5%

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

   if 2.14999999999999996e77 < l

   1. Initial program 31.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 36.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 54.9%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. associate-*r/ N/A

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

       6. associate-*l/ N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 61.4%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(U \cdot 2\right)\right) \cdot \left(\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.3 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(U* - U, \frac{l\_m \cdot n}{Om}, l\_m \cdot -2\right), \frac{l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.3e+144)
   (sqrt
    (*
     U
     (*
      (* n 2.0)
      (fma (fma (- U* U) (/ (* l_m n) Om) (* l_m -2.0)) (/ l_m Om) t))))
   (*
    (sqrt (/ (* (* U n) (fma n (/ (- U* U) Om) -2.0)) Om))
    (* l_m (sqrt 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.3e+144) {
		tmp = sqrt((U * ((n * 2.0) * fma(fma((U_42_ - U), ((l_m * n) / Om), (l_m * -2.0)), (l_m / Om), t))));
	} else {
		tmp = sqrt((((U * n) * fma(n, ((U_42_ - U) / Om), -2.0)) / Om)) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4.3e+144)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(fma(Float64(U_42_ - U), Float64(Float64(l_m * n) / Om), Float64(l_m * -2.0)), Float64(l_m / Om), t))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(U * n) * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)) / Om)) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 4.3e+144], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] / Om), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.29999999999999984e144

   1. Initial program 47.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 59.4%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot -2\right) \cdot \frac{\ell}{Om}} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \ell \cdot -2, \frac{\ell}{Om}, t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right)}, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      4. associate--r- N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(0 - U\right) + U*}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. --lowering--.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 58.8

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

   8. Applied egg-rr 58.8%

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

   if 4.29999999999999984e144 < l

   1. Initial program 30.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 38.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 53.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 61.6%

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

   10. Taylor expanded in l around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sqrt-lowering-sqrt.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. sub-neg N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. /-lowering-/.f64 N/A

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

       12. --lowering--.f64 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. sqrt-lowering-sqrt.f64 67.7

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

   12. Simplified 67.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(U* - U, \frac{\ell \cdot n}{Om}, \ell \cdot -2\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right)\right)}\\ \mathbf{if}\;Om \leq -4.4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(l\_m \cdot n\right) \cdot \frac{U* \cdot \left(l\_m \cdot n\right)}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (* l_m -2.0) t))))))
   (if (<= Om -4.4e+22)
     t_1
     (if (<= Om 6.6e-36)
       (sqrt (/ (* 2.0 (* U (* (* l_m n) (/ (* U* (* l_m n)) Om)))) Om))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U * ((n * 2.0) * fma((l_m / Om), (l_m * -2.0), t))));
	double tmp;
	if (Om <= -4.4e+22) {
		tmp = t_1;
	} else if (Om <= 6.6e-36) {
		tmp = sqrt(((2.0 * (U * ((l_m * n) * ((U_42_ * (l_m * n)) / Om)))) / Om));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(l_m * -2.0), t))))
	tmp = 0.0
	if (Om <= -4.4e+22)
		tmp = t_1;
	elseif (Om <= 6.6e-36)
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(l_m * n) * Float64(Float64(U_42_ * Float64(l_m * n)) / Om)))) / Om));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -4.4e+22], t$95$1, If[LessEqual[Om, 6.6e-36], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(l$95$m * n), $MachinePrecision] * N[(N[(U$42$ * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.4e22 or 6.59999999999999981e-36 < Om

   1. Initial program 52.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 58.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 63.7%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 58.5

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

   9. Simplified 58.5%

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

   if -4.4e22 < Om < 6.59999999999999981e-36

   1. Initial program 36.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 38.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 51.8%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 44.5%

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

   10. Taylor expanded in U* around inf

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-lowering-*.f64 45.8

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

   12. Simplified 45.8%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.4 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \mathbf{elif}\;Om \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(l\_m \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(l\_m \cdot n\right) \cdot \left(U \cdot 2\right)\right) \cdot \left(l\_m \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8.5e+52)
   (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (/ (* U* (* l_m n)) Om) t))))
   (sqrt
    (/ (* (* (* l_m n) (* U 2.0)) (* l_m (fma n (/ (- U* U) Om) -2.0))) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8.5e+52) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), ((U_42_ * (l_m * n)) / Om), t))));
	} else {
		tmp = sqrt(((((l_m * n) * (U * 2.0)) * (l_m * fma(n, ((U_42_ - U) / Om), -2.0))) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8.5e+52)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(l_m * n)) / Om), t))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * n) * Float64(U * 2.0)) * Float64(l_m * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 8.5e+52], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.49999999999999994e52

   1. Initial program 48.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 53.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 60.3%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U* around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 50.7

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

   9. Simplified 50.7%

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

   if 8.49999999999999994e52 < l

   1. Initial program 32.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 37.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 52.6%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. associate-*r/ N/A

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

       6. associate-*l/ N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 58.8%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(U \cdot 2\right)\right) \cdot \left(\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(l\_m, \frac{l\_m \cdot -2}{Om}, t\right)} \cdot \sqrt{U \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -2e-311)
   (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (* l_m -2.0) t))))
   (* (sqrt (* n (fma l_m (/ (* l_m -2.0) Om) t))) (sqrt (* U 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2e-311) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), (l_m * -2.0), t))));
	} else {
		tmp = sqrt((n * fma(l_m, ((l_m * -2.0) / Om), t))) * sqrt((U * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -2e-311)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(l_m * -2.0), t))));
	else
		tmp = Float64(sqrt(Float64(n * fma(l_m, Float64(Float64(l_m * -2.0) / Om), t))) * sqrt(Float64(U * 2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -2e-311], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(l$95$m * N[(N[(l$95$m * -2.0), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1.9999999999999e-311

   1. Initial program 48.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 50.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 59.7%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 47.6

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

   9. Simplified 47.6%

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

   if -1.9999999999999e-311 < U

   1. Initial program 43.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 50.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 58.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 43.8

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

   9. Simplified 43.8%

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

       1. pow1/2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 53.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(\ell, \frac{\ell \cdot -2}{Om}, t\right)} \cdot \sqrt{U \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -4e-311)
   (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (* l_m -2.0) t))))
   (* (sqrt n) (sqrt (* (* U 2.0) (fma -2.0 (/ (* l_m l_m) Om) t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -4e-311) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), (l_m * -2.0), t))));
	} else {
		tmp = sqrt(n) * sqrt(((U * 2.0) * fma(-2.0, ((l_m * l_m) / Om), t)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -4e-311)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(l_m * -2.0), t))));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(U * 2.0) * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -4e-311], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.99999999999979e-311

   1. Initial program 48.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 52.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 60.8%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 48.3

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

   9. Simplified 48.3%

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

   if -3.99999999999979e-311 < n

   1. Initial program 42.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 45.2%

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

   5. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 48.1

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

   7. Simplified 48.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;l\_m \leq 1.7 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(l\_m \cdot -2\right) \cdot \left(l\_m \cdot n\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.3e-74)
   (sqrt (* (* U 2.0) (* n t)))
   (if (<= l_m 1.7e+130)
     (sqrt (* (* U (* n 2.0)) (fma -2.0 (/ (* l_m l_m) Om) t)))
     (sqrt (/ (* 2.0 (* U (* (* l_m -2.0) (* l_m n)))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.3e-74) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else if (l_m <= 1.7e+130) {
		tmp = sqrt(((U * (n * 2.0)) * fma(-2.0, ((l_m * l_m) / Om), t)));
	} else {
		tmp = sqrt(((2.0 * (U * ((l_m * -2.0) * (l_m * n)))) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.3e-74)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	elseif (l_m <= 1.7e+130)
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(l_m * -2.0) * Float64(l_m * n)))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.3e-74], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.7e+130], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.2999999999999998e-74

   1. Initial program 46.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 38.4

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

   5. Simplified 38.4%

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

   if 2.2999999999999998e-74 < l < 1.7e130

   1. Initial program 52.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 58.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.3

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

   7. Simplified 38.3%

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

   if 1.7e130 < l

   1. Initial program 28.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 36.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 53.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 60.8%

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

   10. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 54.1

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

   12. Simplified 54.1%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot -2\right) \cdot \left(\ell \cdot n\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.45 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(l\_m \cdot -2\right) \cdot \left(l\_m \cdot n\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.45e+120)
   (sqrt (* U (* (* n 2.0) (fma (/ l_m Om) (* l_m -2.0) t))))
   (sqrt (/ (* 2.0 (* U (* (* l_m -2.0) (* l_m n)))) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.45e+120) {
		tmp = sqrt((U * ((n * 2.0) * fma((l_m / Om), (l_m * -2.0), t))));
	} else {
		tmp = sqrt(((2.0 * (U * ((l_m * -2.0) * (l_m * n)))) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.45e+120)
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l_m / Om), Float64(l_m * -2.0), t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(l_m * -2.0) * Float64(l_m * n)))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.45e+120], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.45000000000000005e120

   1. Initial program 48.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 52.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 60.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 46.2

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

   9. Simplified 46.2%

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

   if 2.45000000000000005e120 < l

   1. Initial program 27.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 33.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 50.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 57.2%

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

   10. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 50.8

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

   12. Simplified 50.8%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot -2\right) \cdot \left(\ell \cdot n\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.4 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(l\_m \cdot -2\right) \cdot \left(l\_m \cdot n\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.4e+52)
   (sqrt (* (* U 2.0) (* n t)))
   (sqrt (/ (* 2.0 (* U (* (* l_m -2.0) (* l_m n)))) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.4e+52) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = sqrt(((2.0 * (U * ((l_m * -2.0) * (l_m * n)))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 4.4d+52) then
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    else
        tmp = sqrt(((2.0d0 * (u * ((l_m * (-2.0d0)) * (l_m * n)))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.4e+52) {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = Math.sqrt(((2.0 * (U * ((l_m * -2.0) * (l_m * n)))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 4.4e+52:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	else:
		tmp = math.sqrt(((2.0 * (U * ((l_m * -2.0) * (l_m * n)))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4.4e+52)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(l_m * -2.0) * Float64(l_m * n)))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 4.4e+52)
		tmp = sqrt(((U * 2.0) * (n * t)));
	else
		tmp = sqrt(((2.0 * (U * ((l_m * -2.0) * (l_m * n)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 4.4e+52], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e52

   1. Initial program 48.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 37.9

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

   5. Simplified 37.9%

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

   if 4.4e52 < l

   1. Initial program 32.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 37.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   9. Simplified 52.6%

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

   10. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 45.9

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

   12. Simplified 45.9%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot -2\right) \cdot \left(\ell \cdot n\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot -4\right) \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.5e+52)
   (sqrt (* (* U 2.0) (* n t)))
   (sqrt (/ (* (* U -4.0) (* n (* l_m l_m))) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.5e+52) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 3.5d+52) then
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    else
        tmp = sqrt((((u * (-4.0d0)) * (n * (l_m * l_m))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.5e+52) {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = Math.sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 3.5e+52:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	else:
		tmp = math.sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.5e+52)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	else
		tmp = sqrt(Float64(Float64(Float64(U * -4.0) * Float64(n * Float64(l_m * l_m))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 3.5e+52)
		tmp = sqrt(((U * 2.0) * (n * t)));
	else
		tmp = sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.5e+52], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * -4.0), $MachinePrecision] * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e52

   1. Initial program 48.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 37.9

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

   5. Simplified 37.9%

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

   if 3.5e52 < l

   1. Initial program 32.8%

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

   3. Taylor expanded in Om around 0

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

      1. /-lowering-/.f64 N/A

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

   5. Simplified 25.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(Om, \mathsf{fma}\left(Om, t, \left(\ell \cdot \ell\right) \cdot -2\right), \left(n \cdot \left(U - U*\right)\right) \cdot \left(0 - \ell \cdot \ell\right)\right)}{Om \cdot Om}}} \]

   6. Taylor expanded in l around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   8. Simplified 39.4%

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

   9. Taylor expanded in n around 0

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 31.1

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

   11. Simplified 31.1%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot -4\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.6 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \frac{-4 \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.6e+52)
   (sqrt (* (* U 2.0) (* n t)))
   (sqrt (* U (/ (* -4.0 (* n (* l_m l_m))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.6e+52) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = sqrt((U * ((-4.0 * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 3.6d+52) then
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    else
        tmp = sqrt((u * (((-4.0d0) * (n * (l_m * l_m))) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.6e+52) {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = Math.sqrt((U * ((-4.0 * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 3.6e+52:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	else:
		tmp = math.sqrt((U * ((-4.0 * (n * (l_m * l_m))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.6e+52)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	else
		tmp = sqrt(Float64(U * Float64(Float64(-4.0 * Float64(n * Float64(l_m * l_m))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 3.6e+52)
		tmp = sqrt(((U * 2.0) * (n * t)));
	else
		tmp = sqrt((U * ((-4.0 * (n * (l_m * l_m))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.6e+52], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(-4.0 * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.6e52

   1. Initial program 48.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 37.9

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

   5. Simplified 37.9%

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

   if 3.6e52 < l

   1. Initial program 32.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(0 - \left(U - U*\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. metadata-eval N/A

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

      12. --lowering--.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\color{blue}{0} - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \color{blue}{\left(U - U*\right)}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \color{blue}{\frac{\ell}{Om}}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell}{Om}}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

   4. Applied egg-rr 37.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(0 - \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om} + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   6. Applied egg-rr 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(0 - \left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 45.8

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

   9. Simplified 45.8%

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

   10. Taylor expanded in l around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 31.2

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

   12. Simplified 31.2%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \frac{-4 \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -2e-311)
   (sqrt (* (* U 2.0) (* n t)))
   (* (sqrt (* U 2.0)) (sqrt (* n t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2e-311) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = sqrt((U * 2.0)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-2d-311)) then
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    else
        tmp = sqrt((u * 2.0d0)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2e-311) {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = Math.sqrt((U * 2.0)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -2e-311:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	else:
		tmp = math.sqrt((U * 2.0)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -2e-311)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	else
		tmp = Float64(sqrt(Float64(U * 2.0)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -2e-311)
		tmp = sqrt(((U * 2.0) * (n * t)));
	else
		tmp = sqrt((U * 2.0)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -2e-311], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1.9999999999999e-311

   1. Initial program 48.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 40.1

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

   5. Simplified 40.1%

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

   if -1.9999999999999e-311 < U

   1. Initial program 43.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 28.0

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

   5. Simplified 28.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. *-lowering-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 36.2

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

   7. Applied egg-rr 36.2%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -4e-310)
   (sqrt (* (* U 2.0) (* n t)))
   (* (sqrt (* U (* n 2.0))) (sqrt t))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -4e-310) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = sqrt((U * (n * 2.0))) * sqrt(t);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-4d-310)) then
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    else
        tmp = sqrt((u * (n * 2.0d0))) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -4e-310) {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = Math.sqrt((U * (n * 2.0))) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -4e-310:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	else:
		tmp = math.sqrt((U * (n * 2.0))) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -4e-310)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * 2.0))) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -4e-310)
		tmp = sqrt(((U * 2.0) * (n * t)));
	else
		tmp = sqrt((U * (n * 2.0))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -4e-310], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.999999999999988e-310

   1. Initial program 46.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

   if -3.999999999999988e-310 < t

   1. Initial program 45.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 33.7

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

   5. Simplified 33.7%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-lowering-sqrt.f64 40.3

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

   7. Applied egg-rr 40.3%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -5e-73) (sqrt (* (* U 2.0) (* n t))) (sqrt (* n (* t (* U 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -5e-73) {
		tmp = sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = sqrt((n * (t * (U * 2.0))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-5d-73)) then
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    else
        tmp = sqrt((n * (t * (u * 2.0d0))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -5e-73) {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	} else {
		tmp = Math.sqrt((n * (t * (U * 2.0))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -5e-73:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	else:
		tmp = math.sqrt((n * (t * (U * 2.0))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -5e-73)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	else
		tmp = sqrt(Float64(n * Float64(t * Float64(U * 2.0))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -5e-73)
		tmp = sqrt(((U * 2.0) * (n * t)));
	else
		tmp = sqrt((n * (t * (U * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -5e-73], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(t * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -4.9999999999999998e-73

   1. Initial program 58.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 51.2

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

   5. Simplified 51.2%

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

   if -4.9999999999999998e-73 < U

   1. Initial program 41.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 28.5

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

   5. Simplified 28.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 30.7

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

   7. Applied egg-rr 30.7%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Taylor expanded in t around inf

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 33.8

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

5. Simplified 33.8%

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

6. Final simplification 33.8%

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

Reproduce

herbie shell --seed 2024197 
(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*))))))