Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 63.8%

Time: 17.3s

Alternatives: 15

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U - U*\right) \cdot \frac{\ell}{Om}, -n, -2 \cdot \ell\right), t\right)\\ \mathbf{if}\;U \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{t\_1 \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* 2.0 n)
          (fma (/ l Om) (fma (* (- U U*) (/ l Om)) (- n) (* -2.0 l)) t))))
   (if (<= U -4e-310) (sqrt (* t_1 U)) (* (sqrt U) (sqrt t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * fma((l / Om), fma(((U - U_42_) * (l / Om)), -n, (-2.0 * l)), t);
	double tmp;
	if (U <= -4e-310) {
		tmp = sqrt((t_1 * U));
	} else {
		tmp = sqrt(U) * sqrt(t_1);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * fma(Float64(l / Om), fma(Float64(Float64(U - U_42_) * Float64(l / Om)), Float64(-n), Float64(-2.0 * l)), t))
	tmp = 0.0
	if (U <= -4e-310)
		tmp = sqrt(Float64(t_1 * U));
	else
		tmp = Float64(sqrt(U) * sqrt(t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * (-n) + N[(-2.0 * l), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -4e-310], N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -3.999999999999988e-310

   1. Initial program 57.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 64.9

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 62.0%

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

   6. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 71.5%

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

   if -3.999999999999988e-310 < U

   1. Initial program 49.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 54.2

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 50.4%

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

   6. Applied rewrites 49.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 71.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U - U*\right) \cdot \frac{\ell}{Om}, -n, -2 \cdot \ell\right), t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U - U*\right) \cdot \frac{\ell}{Om}, -n, -2 \cdot \ell\right), t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot \ell}{Om} \cdot \frac{\left(\left(U* \cdot U\right) \cdot n\right) \cdot \ell}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_1 2.0) t)) t_2)))
   (if (<= t_3 2e-266)
     (sqrt (* (fma (/ (* (* l l) n) Om) -4.0 (* (* t n) 2.0)) U))
     (if (<= t_3 1e+308)
       (sqrt (* (fma -2.0 t_1 t) t_2))
       (sqrt (* (* (/ (* n l) Om) (/ (* (* (* U* U) n) l) Om)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_1 * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 2e-266) {
		tmp = sqrt((fma((((l * l) * n) / Om), -4.0, ((t * n) * 2.0)) * U));
	} else if (t_3 <= 1e+308) {
		tmp = sqrt((fma(-2.0, t_1, t) * t_2));
	} else {
		tmp = sqrt(((((n * l) / Om) * ((((U_42_ * U) * n) * l) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_1 * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 2e-266)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l * l) * n) / Om), -4.0, Float64(Float64(t * n) * 2.0)) * U));
	elseif (t_3 <= 1e+308)
		tmp = sqrt(Float64(fma(-2.0, t_1, t) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(n * l) / Om) * Float64(Float64(Float64(Float64(U_42_ * U) * n) * l) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-266], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+308], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(N[(U$42$ * U), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2e-266

   1. Initial program 24.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 27.2

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 27.2%

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 45.3

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

   9. Applied rewrites 45.3%

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

   if 2e-266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e308

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. 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. lower-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. lower-/.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. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 23.6%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 10.1

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

   5. Applied rewrites 10.1%

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

   6. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 35.9

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

   8. Applied rewrites 35.9%

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

   10. Applied rewrites 40.1%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot \ell}{Om} \cdot \frac{\left(\left(U* \cdot U\right) \cdot n\right) \cdot \ell}{Om}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(\left(U* \cdot U\right) \cdot n\right) \cdot \ell\right) \cdot n\right) \cdot \ell}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_1 2.0) t)) t_2)))
   (if (<= t_3 2e-266)
     (sqrt (* (fma (/ (* (* l l) n) Om) -4.0 (* (* t n) 2.0)) U))
     (if (<= t_3 1e+308)
       (sqrt (* (fma -2.0 t_1 t) t_2))
       (sqrt (* (/ (* (* (* (* (* U* U) n) l) n) l) (* Om Om)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_1 * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 2e-266) {
		tmp = sqrt((fma((((l * l) * n) / Om), -4.0, ((t * n) * 2.0)) * U));
	} else if (t_3 <= 1e+308) {
		tmp = sqrt((fma(-2.0, t_1, t) * t_2));
	} else {
		tmp = sqrt((((((((U_42_ * U) * n) * l) * n) * l) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_1 * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 2e-266)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l * l) * n) / Om), -4.0, Float64(Float64(t * n) * 2.0)) * U));
	elseif (t_3 <= 1e+308)
		tmp = sqrt(Float64(fma(-2.0, t_1, t) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(U_42_ * U) * n) * l) * n) * l) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-266], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+308], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(N[(U$42$ * U), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2e-266

   1. Initial program 24.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 27.2

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 27.2%

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 45.3

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

   9. Applied rewrites 45.3%

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

   if 2e-266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e308

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. 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. lower-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. lower-/.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. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 23.6%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 10.1

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

   5. Applied rewrites 10.1%

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

   6. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 35.9

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

   8. Applied rewrites 35.9%

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

   10. Applied rewrites 36.7%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(\left(U* \cdot U\right) \cdot n\right) \cdot \ell\right) \cdot n\right) \cdot \ell}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_1 2.0) t)) t_2)))
   (if (<= t_3 2e-266)
     (sqrt (* (fma (/ (* (* l l) n) Om) -4.0 (* (* t n) 2.0)) U))
     (if (<= t_3 1e+308)
       (sqrt (* (fma -2.0 t_1 t) t_2))
       (sqrt (* (/ (* (* (* n l) (* n l)) (* U* U)) (* Om Om)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_1 * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 2e-266) {
		tmp = sqrt((fma((((l * l) * n) / Om), -4.0, ((t * n) * 2.0)) * U));
	} else if (t_3 <= 1e+308) {
		tmp = sqrt((fma(-2.0, t_1, t) * t_2));
	} else {
		tmp = sqrt((((((n * l) * (n * l)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_1 * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 2e-266)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l * l) * n) / Om), -4.0, Float64(Float64(t * n) * 2.0)) * U));
	elseif (t_3 <= 1e+308)
		tmp = sqrt(Float64(fma(-2.0, t_1, t) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * l) * Float64(n * l)) * Float64(U_42_ * U)) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-266], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+308], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2e-266

   1. Initial program 24.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 27.2

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 27.2%

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 45.3

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

   9. Applied rewrites 45.3%

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

   if 2e-266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e308

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. 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. lower-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. lower-/.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. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 23.6%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 10.1

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

   5. Applied rewrites 10.1%

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

   6. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 35.9

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

   8. Applied rewrites 35.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot \ell}{Om \cdot Om} \cdot \left(\left(\left(U* \cdot U\right) \cdot n\right) \cdot \ell\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_1 2.0) t)) t_2)))
   (if (<= t_3 2e-266)
     (sqrt (* (fma (/ (* (* l l) n) Om) -4.0 (* (* t n) 2.0)) U))
     (if (<= t_3 1e+308)
       (sqrt (* (fma -2.0 t_1 t) t_2))
       (sqrt (* (* (/ (* n l) (* Om Om)) (* (* (* U* U) n) l)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_1 * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 2e-266) {
		tmp = sqrt((fma((((l * l) * n) / Om), -4.0, ((t * n) * 2.0)) * U));
	} else if (t_3 <= 1e+308) {
		tmp = sqrt((fma(-2.0, t_1, t) * t_2));
	} else {
		tmp = sqrt(((((n * l) / (Om * Om)) * (((U_42_ * U) * n) * l)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_1 * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 2e-266)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l * l) * n) / Om), -4.0, Float64(Float64(t * n) * 2.0)) * U));
	elseif (t_3 <= 1e+308)
		tmp = sqrt(Float64(fma(-2.0, t_1, t) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(n * l) / Float64(Om * Om)) * Float64(Float64(Float64(U_42_ * U) * n) * l)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-266], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+308], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(n * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U$42$ * U), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2e-266

   1. Initial program 24.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 27.2

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 27.2%

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 45.3

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

   9. Applied rewrites 45.3%

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

   if 2e-266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e308

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. 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. lower-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. lower-/.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. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 23.6%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 10.1

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

   5. Applied rewrites 10.1%

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

   6. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 35.9

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

   8. Applied rewrites 35.9%

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

   10. Applied rewrites 35.1%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, -4, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+308}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{n \cdot \ell}{Om \cdot Om} \cdot \left(\left(\left(U* \cdot U\right) \cdot n\right) \cdot \ell\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (*
         (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* (/ (* l l) Om) 2.0) t))
         t_1)
        0.0)
     (sqrt (* (* t U) (* 2.0 n)))
     (sqrt (* t_1 t)))))

C

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

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (((((u_42 - u) * (((l / om) ** 2.0d0) * n)) - ((((l * l) / om) * 2.0d0) - t)) * t_1) <= 0.0d0) then
        tmp = sqrt(((t * u) * (2.0d0 * n)))
    else
        tmp = sqrt((t_1 * t))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 14.6%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 34.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 58.0%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.3

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

   5. Applied rewrites 36.3%

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

   7. Applied rewrites 37.1%

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

4. Final simplification 36.8%

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

Alternative 7: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U - U*\right) \cdot \frac{\ell}{Om}, -n, -2 \cdot \ell\right), t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\left(-n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.8e+121)
   (sqrt
    (*
     (*
      (* 2.0 n)
      (fma (/ l Om) (fma (* (- U U*) (/ l Om)) (- n) (* -2.0 l)) t))
     U))
   (*
    (* (sqrt 2.0) l)
    (sqrt (* (* (- n) U) (fma (/ (- U U*) Om) (/ n Om) (/ 2.0 Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.8e+121) {
		tmp = sqrt((((2.0 * n) * fma((l / Om), fma(((U - U_42_) * (l / Om)), -n, (-2.0 * l)), t)) * U));
	} else {
		tmp = (sqrt(2.0) * l) * sqrt(((-n * U) * fma(((U - U_42_) / Om), (n / Om), (2.0 / Om))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.8e+121)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * fma(Float64(l / Om), fma(Float64(Float64(U - U_42_) * Float64(l / Om)), Float64(-n), Float64(-2.0 * l)), t)) * U));
	else
		tmp = Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(Float64(Float64(-n) * U) * fma(Float64(Float64(U - U_42_) / Om), Float64(n / Om), Float64(2.0 / Om)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.8e+121], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * (-n) + N[(-2.0 * l), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(N[((-n) * U), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.79999999999999967e121

   1. Initial program 57.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 61.5

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 67.2%

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

   if 7.79999999999999967e121 < l

   1. Initial program 24.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 42.2

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 42.1%

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

   6. Applied rewrites 38.4%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 74.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U - U*\right) \cdot \frac{\ell}{Om}, -n, -2 \cdot \ell\right), t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\left(-n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+51}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\mathsf{fma}\left(-n, U - U*, -2 \cdot Om\right)}{Om} \cdot \frac{\ell}{Om}\right) \cdot \ell\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e+51)
   (sqrt (* (* (* (fma -2.0 (/ (* l l) Om) t) n) U) 2.0))
   (sqrt
    (*
     (* (* (* (/ (fma (- n) (- U U*) (* -2.0 Om)) Om) (/ l Om)) l) U)
     (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e+51) {
		tmp = sqrt((((fma(-2.0, ((l * l) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((fma(-n, (U - U_42_), (-2.0 * Om)) / Om) * (l / Om)) * l) * U) * (2.0 * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e+51)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(fma(Float64(-n), Float64(U - U_42_), Float64(-2.0 * Om)) / Om) * Float64(l / Om)) * l) * U) * Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e+51], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[((-n) * N[(U - U$42$), $MachinePrecision] + N[(-2.0 * Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1e51

   1. Initial program 56.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if 1e51 < l

   1. Initial program 37.2%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 35.4

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

   5. Applied rewrites 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 58.4%

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

   9. Applied rewrites 58.6%

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

Alternative 9: 58.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lift-/.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower--.f64 58.9

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

   18. lift-*.f64 N/A

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

4. Applied rewrites 55.6%

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

6. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

8. Applied rewrites 65.0%

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

9. Final simplification 65.0%

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

Alternative 10: 44.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.5e+105)
   (sqrt (* (* (* (fma -2.0 (/ (* l l) Om) t) n) U) 2.0))
   (sqrt (* (* (* (* (/ l Om) l) -2.0) U) (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.5e+105) {
		tmp = sqrt((((fma(-2.0, ((l * l) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((l / Om) * l) * -2.0) * U) * (2.0 * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.5e+105)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l / Om) * l) * -2.0) * U) * Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.5e+105], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0), $MachinePrecision] * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000023e105

   1. Initial program 57.4%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if 2.50000000000000023e105 < l

   1. Initial program 28.0%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 36.2

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

   5. Applied rewrites 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 60.0%

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

   9. Applied rewrites 60.1%

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

   10. Taylor expanded in Om around inf

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

   12. Applied rewrites 33.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.9e+82)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (* (* (* (/ l Om) l) -2.0) U) (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.9e+82) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((l / Om) * l) * -2.0) * U) * (2.0 * n)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 2.9d+82) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((((((l / om) * l) * (-2.0d0)) * u) * (2.0d0 * n)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.9e+82) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((((l / Om) * l) * -2.0) * U) * (2.0 * n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.9e+82:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((((((l / Om) * l) * -2.0) * U) * (2.0 * n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.9e+82)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l / Om) * l) * -2.0) * U) * Float64(2.0 * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.9e+82)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((((((l / Om) * l) * -2.0) * U) * (2.0 * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.9e+82], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0), $MachinePrecision] * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9000000000000001e82

   1. Initial program 57.2%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 41.9

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

   5. Applied rewrites 41.9%

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

   if 2.9000000000000001e82 < l

   1. Initial program 30.9%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 33.7

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

   5. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in Om around inf

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

   12. Applied rewrites 33.4%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\ell \cdot \ell}{Om} \cdot -2\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.9e+82)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (* (* (/ (* l l) Om) -2.0) U) (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.9e+82) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((l * l) / Om) * -2.0) * U) * (2.0 * n)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 2.9d+82) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((((((l * l) / om) * (-2.0d0)) * u) * (2.0d0 * n)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.9e+82) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((((l * l) / Om) * -2.0) * U) * (2.0 * n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.9e+82:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((((((l * l) / Om) * -2.0) * U) * (2.0 * n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.9e+82)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / Om) * -2.0) * U) * Float64(2.0 * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.9e+82)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((((((l * l) / Om) * -2.0) * U) * (2.0 * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.9e+82], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision] * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9000000000000001e82

   1. Initial program 57.2%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 41.9

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

   5. Applied rewrites 41.9%

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

   if 2.9000000000000001e82 < l

   1. Initial program 30.9%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 33.7

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

   5. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in Om around inf

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

   10. Applied rewrites 22.4%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\ell \cdot \ell}{Om} \cdot -2\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot U} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 2.5e-75)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* t U)) (sqrt (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 2.5e-75) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((t * U)) * sqrt((2.0 * n));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (n <= 2.5d-75) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((t * u)) * sqrt((2.0d0 * n))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 2.5e-75) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((t * U)) * Math.sqrt((2.0 * n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 2.5e-75:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((t * U)) * math.sqrt((2.0 * n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 2.5e-75)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(t * U)) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 2.5e-75)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((t * U)) * sqrt((2.0 * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 2.5e-75], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.49999999999999989e-75

   1. Initial program 55.4%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if 2.49999999999999989e-75 < n

   1. Initial program 46.8%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 25.3

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

   5. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 47.0%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 14.2

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

   10. Applied rewrites 14.2%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. lower-*.f64 N/A

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

       7. pow1/2 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. pow1/2 N/A

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

       10. lower-sqrt.f64 24.5

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

   12. Applied rewrites 24.5%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot U} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

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((((t * n) * u) * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 36.1

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

5. Applied rewrites 36.1%

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

6. Final simplification 36.1%

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

Alternative 15: 35.7% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

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))
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));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 36.1

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

5. Applied rewrites 36.1%

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

7. Applied rewrites 35.0%

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

8. Final simplification 35.0%

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

Reproduce

herbie shell --seed 2024241 
(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*))))))