Toniolo and Linder, Equation (13)

Percentage Accurate: 50.4% → 65.7%

Time: 13.0s

Alternatives: 20

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% 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: 65.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(n \cdot \frac{\ell}{Om}, U* - U, -2 \cdot \ell\right), t\right)\\ \mathbf{if}\;U \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(t\_1 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ l Om) (fma (* n (/ l Om)) (- U* U) (* -2.0 l)) t)))
   (if (<= U -1e-310)
     (sqrt (* (* 2.0 n) (* t_1 U)))
     (* (sqrt U) (sqrt (* (* 2.0 n) t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((l / Om), fma((n * (l / Om)), (U_42_ - U), (-2.0 * l)), t);
	double tmp;
	if (U <= -1e-310) {
		tmp = sqrt(((2.0 * n) * (t_1 * U)));
	} else {
		tmp = sqrt(U) * sqrt(((2.0 * n) * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(l / Om), fma(Float64(n * Float64(l / Om)), Float64(U_42_ - U), Float64(-2.0 * l)), t)
	tmp = 0.0
	if (U <= -1e-310)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(t_1 * U)));
	else
		tmp = Float64(sqrt(U) * sqrt(Float64(Float64(2.0 * n) * t_1)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if U < -9.999999999999969e-311

   1. Initial program 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 54.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 56.9

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 56.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if -9.999999999999969e-311 < U

   1. Initial program 43.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 49.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 50.3

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 50.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 65.7%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(n \cdot \frac{\ell}{Om}, U* - U, -2 \cdot \ell\right), t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(n \cdot \frac{\ell}{Om}, U* - U, -2 \cdot \ell\right), t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{\left(\left(U* \cdot U\right) \cdot n\right) \cdot n}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (sqrt (* (* (* (/ l Om) l) (/ (* (* (* U* U) n) n) Om)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = sqrt(((((l / Om) * l) * ((((U_42_ * U) * n) * n) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l / Om) * l) * Float64(Float64(Float64(Float64(U_42_ * U) * n) * n) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(U$42$ * U), $MachinePrecision] * n), $MachinePrecision] * n), $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*)))) < 0.0

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

   3. Taylor expanded in U* around inf

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

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

   7. Applied rewrites 31.7%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \frac{\left(\left(U* \cdot U\right) \cdot n\right) \cdot n}{Om}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot \left(U* \cdot U\right)\right) \cdot \left(n \cdot \ell\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (sqrt (* (/ (* (* (* n l) (* U* U)) (* 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 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = sqrt((((((n * l) * (U_42_ * U)) * (n * l)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * l) * Float64(U_42_ * U)) * Float64(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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * l), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] * N[(n * l), $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*)))) < 0.0

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

   3. Taylor expanded in U* around inf

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

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

   7. Applied rewrites 31.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot \left(U* \cdot U\right)\right) \cdot \left(n \cdot \ell\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (sqrt (* (/ (* (* (* n n) (* l 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 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = sqrt((((((n * n) * (l * l)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * n) * Float64(l * 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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * n), $MachinePrecision] * N[(l * 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*)))) < 0.0

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

   3. Taylor expanded in U* around inf

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

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot n}{Om} \cdot \left(\sqrt{U* \cdot U} \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (* (/ (* (sqrt 2.0) n) Om) (* (sqrt (* U* U)) l))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = ((sqrt(2.0) * n) / Om) * (sqrt((U_42_ * U)) * l);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) * n) / Om) * Float64(sqrt(Float64(U_42_ * U)) * l));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 28.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 16.0

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

   9. Applied rewrites 16.0%

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

   11. Applied rewrites 19.8%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot n}{Om} \cdot \left(\sqrt{U* \cdot U} \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2} \cdot n}{Om} \cdot \sqrt{U* \cdot U}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (* (* (/ (* (sqrt 2.0) n) Om) (sqrt (* U* U))) l)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = (((sqrt(2.0) * n) / Om) * sqrt((U_42_ * U))) * l;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * n) / Om) * sqrt(Float64(U_42_ * U))) * l);
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 28.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 16.0

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

   9. Applied rewrites 16.0%

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

   11. Applied rewrites 20.6%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2} \cdot n}{Om} \cdot \sqrt{U* \cdot U}\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{U* \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (* (/ (* (* (sqrt 2.0) n) l) Om) (sqrt (* U* U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = (((sqrt(2.0) * n) * l) / Om) * sqrt((U_42_ * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om) * sqrt(Float64(U_42_ * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 16.0

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

   5. Applied rewrites 16.0%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{U* \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(n \cdot \ell\right)}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (* l l) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*))) t_1)))
   (if (<= t_3 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+293)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (/ (* (sqrt (* (* U* U) 2.0)) (* n l)) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = (l * l) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+293) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = (sqrt(((U_42_ * U) * 2.0)) * (n * l)) / Om;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+293)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) * Float64(n * l)) / Om);
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+293], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 28.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 16.0

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

   9. Applied rewrites 16.0%

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

   11. Applied rewrites 15.2%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(n \cdot \ell\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.1% accurate, 0.5× speedup

Math

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

FPCore

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

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(Float64(t - Float64(t_1 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * Float64(Float64(2.0 * n) * U))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(fma(-2.0, t_1, t) * U) * Float64(2.0 * n)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) * Float64(n * l)) / Om);
	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[(N[(t - N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 66.3%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.9

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

   9. Applied rewrites 52.9%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 0.4

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

   4. Applied rewrites 0.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 8.2%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 11.7

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

   9. Applied rewrites 11.7%

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

   11. Applied rewrites 11.7%

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

4. Final simplification 45.0%

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

Alternative 10: 47.0% accurate, 0.5× speedup

Math

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

FPCore

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

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(Float64(t - Float64(t_1 * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * Float64(Float64(2.0 * n) * U))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, t_1, t) * n) * U) * 2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) * Float64(n * l)) / Om);
	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[(N[(t - N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 66.8%

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

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

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

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

      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 48.4

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

   5. Applied rewrites 48.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 0.4

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

   4. Applied rewrites 0.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 8.2%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 11.7

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

   9. Applied rewrites 11.7%

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

   11. Applied rewrites 11.7%

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

4. Final simplification 41.9%

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

Alternative 11: 42.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 10^{+150}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(t \cdot n\right) \cdot U\right| \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (-
            (- t (* (/ (* l l) Om) 2.0))
            (* (* (pow (/ l Om) 2.0) n) (- U U*)))
           (* (* 2.0 n) U)))))
   (if (<= t_1 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_1 1e+150)
       (sqrt (* (* (* n U) t) 2.0))
       (sqrt (* (fabs (* (* t n) U)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((t - (((l * l) / Om) * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_1 <= 1e+150) {
		tmp = sqrt((((n * U) * t) * 2.0));
	} else {
		tmp = sqrt((fabs(((t * n) * U)) * 2.0));
	}
	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 = sqrt((((t - (((l * l) / om) * 2.0d0)) - ((((l / om) ** 2.0d0) * n) * (u - u_42))) * ((2.0d0 * n) * u)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * t)) * sqrt(u)
    else if (t_1 <= 1d+150) then
        tmp = sqrt((((n * u) * t) * 2.0d0))
    else
        tmp = sqrt((abs(((t * n) * u)) * 2.0d0))
    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 = Math.sqrt((((t - (((l * l) / Om) * 2.0)) - ((Math.pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * t)) * Math.sqrt(U);
	} else if (t_1 <= 1e+150) {
		tmp = Math.sqrt((((n * U) * t) * 2.0));
	} else {
		tmp = Math.sqrt((Math.abs(((t * n) * U)) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((t - (((l * l) / Om) * 2.0)) - ((math.pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * t)) * math.sqrt(U)
	elif t_1 <= 1e+150:
		tmp = math.sqrt((((n * U) * t) * 2.0))
	else:
		tmp = math.sqrt((math.fabs(((t * n) * U)) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(t - Float64(Float64(Float64(l * l) / Om) * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * Float64(Float64(2.0 * n) * U)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_1 <= 1e+150)
		tmp = sqrt(Float64(Float64(Float64(n * U) * t) * 2.0));
	else
		tmp = sqrt(Float64(abs(Float64(Float64(t * n) * U)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((t - (((l * l) / Om) * 2.0)) - ((((l / Om) ^ 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	elseif (t_1 <= 1e+150)
		tmp = sqrt((((n * U) * t) * 2.0));
	else
		tmp = sqrt((abs(((t * n) * U)) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(t - N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+150], N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[Abs[N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.7%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 47.9

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

   7. Applied rewrites 47.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.99999999999999981e149

   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 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. *-commutative N/A

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

      6. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   7. Applied rewrites 72.6%

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

   if 9.99999999999999981e149 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 19.2%

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

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

      6. lower-*.f64 10.6

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

   5. Applied rewrites 10.6%

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

   7. Applied rewrites 16.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq 10^{+150}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(t \cdot n\right) \cdot U\right| \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(n \cdot \ell\right)}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (-
           (- t (* (/ (* l l) Om) 2.0))
           (* (* (pow (/ l Om) 2.0) n) (- U U*)))
          (* (* 2.0 n) U))))
   (if (<= t_1 0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_1 1e+293)
       (sqrt (* (* (* n U) t) 2.0))
       (/ (* (sqrt (* (* U* U) 2.0)) (* n l)) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((t - (((l * l) / Om) * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_1 <= 1e+293) {
		tmp = sqrt((((n * U) * t) * 2.0));
	} else {
		tmp = (sqrt(((U_42_ * U) * 2.0)) * (n * l)) / Om;
	}
	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 = ((t - (((l * l) / om) * 2.0d0)) - ((((l / om) ** 2.0d0) * n) * (u - u_42))) * ((2.0d0 * n) * u)
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * t)) * sqrt(u)
    else if (t_1 <= 1d+293) then
        tmp = sqrt((((n * u) * t) * 2.0d0))
    else
        tmp = (sqrt(((u_42 * u) * 2.0d0)) * (n * l)) / om
    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 = ((t - (((l * l) / Om) * 2.0)) - ((Math.pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * t)) * Math.sqrt(U);
	} else if (t_1 <= 1e+293) {
		tmp = Math.sqrt((((n * U) * t) * 2.0));
	} else {
		tmp = (Math.sqrt(((U_42_ * U) * 2.0)) * (n * l)) / Om;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((t - (((l * l) / Om) * 2.0)) - ((math.pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U)
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * t)) * math.sqrt(U)
	elif t_1 <= 1e+293:
		tmp = math.sqrt((((n * U) * t) * 2.0))
	else:
		tmp = (math.sqrt(((U_42_ * U) * 2.0)) * (n * l)) / Om
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(t - Float64(Float64(Float64(l * l) / Om) * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * Float64(Float64(2.0 * n) * U))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_1 <= 1e+293)
		tmp = sqrt(Float64(Float64(Float64(n * U) * t) * 2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) * Float64(n * l)) / Om);
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((t - (((l * l) / Om) * 2.0)) - ((((l / Om) ^ 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	elseif (t_1 <= 1e+293)
		tmp = sqrt((((n * U) * t) * 2.0));
	else
		tmp = (sqrt(((U_42_ * U) * 2.0)) * (n * l)) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(t - N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+293], N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.0%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 44.8

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

   7. Applied rewrites 44.8%

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

   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*)))) < 9.9999999999999992e292

   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 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. *-commutative N/A

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

      6. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 73.3%

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

   if 9.9999999999999992e292 < (*.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 20.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 28.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 16.0

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

   9. Applied rewrites 16.0%

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

   11. Applied rewrites 15.2%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 10^{+293}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(n \cdot \ell\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.7%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 47.9

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

   7. Applied rewrites 47.9%

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

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

   1. Initial program 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 55.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 57.3

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 57.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

4. Final simplification 63.7%

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

Alternative 14: 62.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          (-
           (- t (* (/ (* l l) Om) 2.0))
           (* (* (pow (/ l Om) 2.0) n) (- U U*)))
          t_1))
        0.0)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (sqrt
      (* t_1 (fma (/ l Om) (fma (* n (/ l Om)) (- U* U) (* -2.0 l)) 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 (sqrt((((t - (((l * l) / Om) * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * t_1)) <= 0.0) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else {
		tmp = sqrt((t_1 * fma((l / Om), fma((n * (l / Om)), (U_42_ - U), (-2.0 * l)), 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 (sqrt(Float64(Float64(Float64(t - Float64(Float64(Float64(l * l) / Om) * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * t_1)) <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	else
		tmp = sqrt(Float64(t_1 * fma(Float64(l / Om), fma(Float64(n * Float64(l / Om)), Float64(U_42_ - U), Float64(-2.0 * l)), t)));
	end
	return 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[Sqrt[N[(N[(N[(t - N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 * l), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.7%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 47.9

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

   7. Applied rewrites 47.9%

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

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

   1. Initial program 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 55.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 57.3

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 56.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.3

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. associate-+r+ N/A

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

   10. Applied rewrites 63.7%

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

4. Final simplification 61.3%

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

Alternative 15: 39.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(t \cdot n\right) \cdot U\right| \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (- (- t (* (/ (* l l) Om) 2.0)) (* (* (pow (/ l Om) 2.0) n) (- U U*)))
        (* (* 2.0 n) U)))
      5e+116)
   (sqrt (* (* t U) (* 2.0 n)))
   (sqrt (* (fabs (* (* t n) U)) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (sqrt((((t - (((l * l) / Om) * 2.0)) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U))) <= 5e+116) {
		tmp = sqrt(((t * U) * (2.0 * n)));
	} else {
		tmp = sqrt((fabs(((t * n) * U)) * 2.0));
	}
	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 (sqrt((((t - (((l * l) / om) * 2.0d0)) - ((((l / om) ** 2.0d0) * n) * (u - u_42))) * ((2.0d0 * n) * u))) <= 5d+116) then
        tmp = sqrt(((t * u) * (2.0d0 * n)))
    else
        tmp = sqrt((abs(((t * n) * u)) * 2.0d0))
    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 (Math.sqrt((((t - (((l * l) / Om) * 2.0)) - ((Math.pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U))) <= 5e+116) {
		tmp = Math.sqrt(((t * U) * (2.0 * n)));
	} else {
		tmp = Math.sqrt((Math.abs(((t * n) * U)) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((t - (((l * l) / Om) * 2.0)) - ((math.pow((l / Om), 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U))) <= 5e+116:
		tmp = math.sqrt(((t * U) * (2.0 * n)))
	else:
		tmp = math.sqrt((math.fabs(((t * n) * U)) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(t - Float64(Float64(Float64(l * l) / Om) * 2.0)) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * Float64(Float64(2.0 * n) * U))) <= 5e+116)
		tmp = sqrt(Float64(Float64(t * U) * Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(abs(Float64(Float64(t * n) * U)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((t - (((l * l) / Om) * 2.0)) - ((((l / Om) ^ 2.0) * n) * (U - U_42_))) * ((2.0 * n) * U))) <= 5e+116)
		tmp = sqrt(((t * U) * (2.0 * n)));
	else
		tmp = sqrt((abs(((t * n) * U)) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(t - N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+116], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[Abs[N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000025e116

   1. Initial program 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in t around inf

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

      1. lower-*.f64 58.9

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

   9. Applied rewrites 58.9%

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

   if 5.00000000000000025e116 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 23.4%

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

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

      6. lower-*.f64 13.8

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

   5. Applied rewrites 13.8%

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

   7. Applied rewrites 19.2%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(t \cdot n\right) \cdot U\right| \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.02e-131)
   (sqrt (* (* t U) (* 2.0 n)))
   (sqrt
    (*
     (* (fma (* l l) (- (/ (* (- U* U) n) (* Om Om)) (/ 2.0 Om)) t) U)
     (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.02e-131) {
		tmp = sqrt(((t * U) * (2.0 * n)));
	} else {
		tmp = sqrt(((fma((l * l), ((((U_42_ - U) * n) / (Om * Om)) - (2.0 / Om)), t) * U) * (2.0 * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.02e-131)
		tmp = sqrt(Float64(Float64(t * U) * Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(Float64(fma(Float64(l * l), Float64(Float64(Float64(Float64(U_42_ - U) * n) / Float64(Om * Om)) - Float64(2.0 / Om)), t) * U) * Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.02e-131], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.02000000000000001e-131

   1. Initial program 48.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.0

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

   4. Applied rewrites 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-*.f64 42.3

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

   9. Applied rewrites 42.3%

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

   if 1.02000000000000001e-131 < l

   1. Initial program 45.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 48.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 51.1

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f64 51.8

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

   11. Applied rewrites 51.8%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left|\left(t \cdot n\right) \cdot U\right| \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -5e-310)
   (sqrt (* (fabs (* (* t n) U)) 2.0))
   (* (sqrt (* (* t U) 2.0)) (sqrt n))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -5e-310) {
		tmp = sqrt((fabs(((t * n) * U)) * 2.0));
	} else {
		tmp = sqrt(((t * U) * 2.0)) * sqrt(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 <= (-5d-310)) then
        tmp = sqrt((abs(((t * n) * u)) * 2.0d0))
    else
        tmp = sqrt(((t * u) * 2.0d0)) * sqrt(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 <= -5e-310) {
		tmp = Math.sqrt((Math.abs(((t * n) * U)) * 2.0));
	} else {
		tmp = Math.sqrt(((t * U) * 2.0)) * Math.sqrt(n);
	}
	return tmp;
}

Python

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

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = sqrt(Float64(abs(Float64(Float64(t * n) * U)) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(t * U) * 2.0)) * sqrt(n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -5e-310)
		tmp = sqrt((abs(((t * n) * U)) * 2.0));
	else
		tmp = sqrt(((t * U) * 2.0)) * sqrt(n);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -5e-310], N[Sqrt[N[(N[Abs[N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 42.6%

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

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

      6. lower-*.f64 28.5

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

   5. Applied rewrites 28.5%

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

   7. Applied rewrites 31.7%

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

   if -4.999999999999985e-310 < n

   1. Initial program 51.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 46.6

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

   7. Applied rewrites 46.6%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left|\left(t \cdot n\right) \cdot U\right| \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 51.0

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

4. Applied rewrites 51.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 51.7%

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

7. Taylor expanded in t around inf

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

   1. lower-*.f64 34.6

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

9. Applied rewrites 34.6%

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

10. Final simplification 34.6%

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

Alternative 19: 36.0% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* (* t U) n) 2.0)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((t * U) * n) * 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 * u) * n) * 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 * U) * n) * 2.0));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((t * U) * n) * 2.0))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(t * U) * n) * 2.0))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((t * U) * n) * 2.0));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 47.1%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. 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. *-commutative N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(t \cdot n\right)} \cdot U\right) \cdot 2} \]

   6. lower-*.f64 31.1

      \[\leadsto \sqrt{\left(\color{blue}{\left(t \cdot n\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 31.1%

   \[\leadsto \sqrt{\color{blue}{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 34.6%

   \[\leadsto \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2} \]
8. Add Preprocessing

Alternative 20: 36.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* (* n U) t) 2.0)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((n * U) * t) * 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((((n * u) * t) * 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((((n * U) * t) * 2.0));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((n * U) * t) * 2.0))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(n * U) * t) * 2.0))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((n * U) * t) * 2.0));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 47.1%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. 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. *-commutative N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(t \cdot n\right)} \cdot U\right) \cdot 2} \]

   6. lower-*.f64 31.1

      \[\leadsto \sqrt{\left(\color{blue}{\left(t \cdot n\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 31.1%

   \[\leadsto \sqrt{\color{blue}{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 31.4%

   \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \]

8. Final simplification 31.4%

   \[\leadsto \sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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*))))))