Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 61.5%

Time: 17.7s

Alternatives: 15

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.9% 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: 61.5% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (* (* 2.0 n) (fma (* (/ l Om) l) (- -2.0 (* (/ (- n) Om) U*)) t))))
   (if (<= U -2e-310) (sqrt (* t_1 U)) (* (sqrt U) (sqrt t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * fma(((l / Om) * l), (-2.0 - ((-n / Om) * U_42_)), t);
	double tmp;
	if (U <= -2e-310) {
		tmp = sqrt((t_1 * U));
	} else {
		tmp = sqrt(U) * sqrt(t_1);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * fma(Float64(Float64(l / Om) * l), Float64(-2.0 - Float64(Float64(Float64(-n) / Om) * U_42_)), t))
	tmp = 0.0
	if (U <= -2e-310)
		tmp = sqrt(Float64(t_1 * U));
	else
		tmp = Float64(sqrt(U) * sqrt(t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * N[(-2.0 - N[(N[((-n) / Om), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -2e-310], N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -1.999999999999994e-310

   1. Initial program 56.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.2

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

   3. Applied rewrites 53.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 58.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.2

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

   6. Applied rewrites 58.2%

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

   7. Taylor expanded in U* around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 61.3

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

   9. Applied rewrites 61.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 63.8%

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

   if -1.999999999999994e-310 < U

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.8

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

   3. Applied rewrites 54.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 61.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.1

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

   6. Applied rewrites 61.1%

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

   7. Taylor expanded in U* around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 61.2

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

   9. Applied rewrites 61.2%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. sqrt-prod N/A

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

       7. pow1/2 N/A

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

       8. lower-*.f64 N/A

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

   11. Applied rewrites 75.0%

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

4. Final simplification 69.4%

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

Alternative 2: 51.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.1

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

   3. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 52.8

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 52.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 51.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 59.4%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.0

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

   3. Applied rewrites 40.8%

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

   5. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 41.0

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

   7. Applied rewrites 41.0%

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

4. Final simplification 57.4%

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

Alternative 3: 57.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2 - \frac{-n}{Om} \cdot U*, t\right)\\ \mathbf{if}\;t \leq 7 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot t\_1\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l Om) l) (- -2.0 (* (/ (- n) Om) U*)) t)))
   (if (<= t 7e+219)
     (sqrt (* (* (* 2.0 n) t_1) U))
     (* (sqrt (* n U)) (sqrt (* 2.0 t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(((l / Om) * l), (-2.0 - ((-n / Om) * U_42_)), t);
	double tmp;
	if (t <= 7e+219) {
		tmp = sqrt((((2.0 * n) * t_1) * U));
	} else {
		tmp = sqrt((n * U)) * sqrt((2.0 * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(Float64(l / Om) * l), Float64(-2.0 - Float64(Float64(Float64(-n) / Om) * U_42_)), t)
	tmp = 0.0
	if (t <= 7e+219)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * t_1) * U));
	else
		tmp = Float64(sqrt(Float64(n * U)) * sqrt(Float64(2.0 * t_1)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * N[(-2.0 - N[(N[((-n) / Om), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t, 7e+219], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * t$95$1), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.0000000000000002e219

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.4

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

   3. Applied rewrites 53.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 59.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.8

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

   6. Applied rewrites 59.8%

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

   7. Taylor expanded in U* around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 61.2

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

   9. Applied rewrites 61.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 64.4%

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

   if 7.0000000000000002e219 < t

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.1

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

   3. Applied rewrites 53.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 58.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in U* around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 62.4

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

   9. Applied rewrites 62.4%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. sqrt-prod N/A

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

       11. pow1/2 N/A

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

   11. Applied rewrites 82.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2 - \frac{-n}{Om} \cdot U*, t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2 - \frac{-n}{Om} \cdot U*, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{U* - U}{Om}, n, -2\right) \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (fma (* (fma (/ (- U* U) Om) n -2.0) l) (/ l Om) t)
           (* (* 2.0 n) U)))))
   (if (<= n -1.25e-97)
     t_1
     (if (<= n 1.25e-123)
       (sqrt (* (* (fma (* (/ l Om) l) -2.0 t) (* 2.0 n)) U))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((fma((fma(((U_42_ - U) / Om), n, -2.0) * l), (l / Om), t) * ((2.0 * n) * U)));
	double tmp;
	if (n <= -1.25e-97) {
		tmp = t_1;
	} else if (n <= 1.25e-123) {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) * (2.0 * n)) * U));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(fma(Float64(fma(Float64(Float64(U_42_ - U) / Om), n, -2.0) * l), Float64(l / Om), t) * Float64(Float64(2.0 * n) * U)))
	tmp = 0.0
	if (n <= -1.25e-97)
		tmp = t_1;
	elseif (n <= 1.25e-123)
		tmp = sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(2.0 * n)) * U));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * n + -2.0), $MachinePrecision] * l), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.25e-97], t$95$1, If[LessEqual[n, 1.25e-123], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2499999999999999e-97 or 1.25000000000000007e-123 < n

   1. Initial program 54.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.9

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

   3. Applied rewrites 56.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 64.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.5

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in n around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 64.5

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

   9. Applied rewrites 64.5%

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

   if -1.2499999999999999e-97 < n < 1.25000000000000007e-123

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.8

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

   3. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 44.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 44.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 44.4

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

   6. Applied rewrites 44.4%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 47.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 65.5%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{U* - U}{Om}, n, -2\right) \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{U* - U}{Om}, n, -2\right) \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2 - \frac{-n}{Om} \cdot U*, t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.1e+228)
   (sqrt
    (* (* (* 2.0 n) (fma (* (/ l Om) l) (- -2.0 (* (/ (- n) Om) U*)) t)) U))
   (* (sqrt t) (sqrt (* (* 2.0 n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.1e+228) {
		tmp = sqrt((((2.0 * n) * fma(((l / Om) * l), (-2.0 - ((-n / Om) * U_42_)), t)) * U));
	} else {
		tmp = sqrt(t) * sqrt(((2.0 * n) * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.1e+228)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * fma(Float64(Float64(l / Om) * l), Float64(-2.0 - Float64(Float64(Float64(-n) / Om) * U_42_)), t)) * U));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(2.0 * n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.1e+228], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * N[(-2.0 - N[(N[((-n) / Om), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e228

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.4

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

   3. Applied rewrites 53.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 59.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in U* around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 61.1

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

   9. Applied rewrites 61.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 64.2%

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

   if 1.1e228 < t

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.3

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.0

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 49.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.0

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

   6. Applied rewrites 49.0%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 59.8%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. unpow-prod-down N/A

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

       10. lower-*.f64 N/A

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

   11. Applied rewrites 84.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2 - \frac{-n}{Om} \cdot U*, t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.0245:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{n \cdot U*}{Om} \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 0.0245)
   (sqrt (* (* (fma (* (/ l Om) l) -2.0 t) (* 2.0 n)) U))
   (sqrt (* (fma (* (/ (* n U*) Om) l) (/ l Om) t) (* (* 2.0 n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 0.0245) {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) * (2.0 * n)) * U));
	} else {
		tmp = sqrt((fma((((n * U_42_) / Om) * l), (l / Om), t) * ((2.0 * n) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 0.0245)
		tmp = sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(2.0 * n)) * U));
	else
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(n * U_42_) / Om) * l), Float64(l / Om), t) * Float64(Float64(2.0 * n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 0.0245], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.024500000000000001

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.4

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

   3. Applied rewrites 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.1

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 49.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 48.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 60.5%

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

   if 0.024500000000000001 < n

   1. Initial program 51.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.2

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

   3. Applied rewrites 57.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 65.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.5

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

   6. Applied rewrites 65.5%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.5

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

   9. Applied rewrites 60.5%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.0245:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{n \cdot U*}{Om} \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;U \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{t\_1 \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l Om) l) -2.0 t) (* 2.0 n))))
   (if (<= U -4.1e-297) (sqrt (* t_1 U)) (* (sqrt t_1) (sqrt U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(((l / Om) * l), -2.0, t) * (2.0 * n);
	double tmp;
	if (U <= -4.1e-297) {
		tmp = sqrt((t_1 * U));
	} else {
		tmp = sqrt(t_1) * sqrt(U);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(2.0 * n))
	tmp = 0.0
	if (U <= -4.1e-297)
		tmp = sqrt(Float64(t_1 * U));
	else
		tmp = Float64(sqrt(t_1) * sqrt(U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -4.1e-297], N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.1000000000000002e-297

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.6

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

   3. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 51.1

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 51.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.1

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 48.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 56.6%

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

   if -4.1000000000000002e-297 < U

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.5

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

   3. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 52.8

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 52.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 45.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. sqrt-prod N/A

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

       11. pow1/2 N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 64.0%

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

Alternative 8: 50.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\\ \mathbf{if}\;n \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(t\_1 \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l Om) l) -2.0 t)))
   (if (<= n 1.8e-26)
     (sqrt (* (* t_1 (* 2.0 n)) U))
     (* (sqrt n) (sqrt (* (* t_1 U) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(((l / Om) * l), -2.0, t);
	double tmp;
	if (n <= 1.8e-26) {
		tmp = sqrt(((t_1 * (2.0 * n)) * U));
	} else {
		tmp = sqrt(n) * sqrt(((t_1 * U) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(Float64(l / Om) * l), -2.0, t)
	tmp = 0.0
	if (n <= 1.8e-26)
		tmp = sqrt(Float64(Float64(t_1 * Float64(2.0 * n)) * U));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(t_1 * U) * 2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[n, 1.8e-26], N[Sqrt[N[(N[(t$95$1 * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 1.8000000000000001e-26

   1. Initial program 53.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.7

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

   3. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 48.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 48.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 48.2

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

   6. Applied rewrites 48.2%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 47.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 60.0%

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

   if 1.8000000000000001e-26 < n

   1. Initial program 53.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.0

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

   3. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.0

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 60.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.0

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

   6. Applied rewrites 60.0%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 45.4%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. sqrt-prod N/A

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

       7. pow1/2 N/A

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

       8. lower-*.f64 N/A

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

   11. Applied rewrites 54.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.3 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot \ell\right) \cdot \frac{U* \cdot \ell}{Om \cdot Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 2.3e+119)
   (sqrt (* (* (fma (* (/ l Om) l) -2.0 t) (* 2.0 n)) U))
   (sqrt (* (* (* n l) (/ (* U* l) (* Om Om))) (* (* 2.0 n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 2.3e+119) {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) * (2.0 * n)) * U));
	} else {
		tmp = sqrt((((n * l) * ((U_42_ * l) / (Om * Om))) * ((2.0 * n) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 2.3e+119)
		tmp = sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(2.0 * n)) * U));
	else
		tmp = sqrt(Float64(Float64(Float64(n * l) * Float64(Float64(U_42_ * l) / Float64(Om * Om))) * Float64(Float64(2.0 * n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 2.3e+119], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n * l), $MachinePrecision] * N[(N[(U$42$ * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.3000000000000001e119

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.5

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 51.8

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 51.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.8

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 49.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 58.8%

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

   if 2.3000000000000001e119 < n

   1. Initial program 48.6%

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

   3. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 41.3

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

   5. Applied rewrites 41.3%

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

   7. Applied rewrites 51.1%

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

   9. Applied rewrites 52.6%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.3 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot \ell\right) \cdot \frac{U* \cdot \ell}{Om \cdot Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 6e+227)
   (sqrt (* (* (fma (* (/ l Om) l) -2.0 t) (* 2.0 n)) U))
   (* (sqrt t) (sqrt (* (* 2.0 n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 6e+227) {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) * (2.0 * n)) * U));
	} else {
		tmp = sqrt(t) * sqrt(((2.0 * n) * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 6e+227)
		tmp = sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(2.0 * n)) * U));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(2.0 * n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 6e+227], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.99999999999999972e227

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.4

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

   3. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 52.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 52.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.2

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in Om around inf

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

   9. Applied rewrites 46.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 54.7%

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

   if 5.99999999999999972e227 < t

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.3

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.0

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 49.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.0

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

   6. Applied rewrites 49.0%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 59.8%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. unpow-prod-down N/A

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

       10. lower-*.f64 N/A

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

   11. Applied rewrites 84.8%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2.8e+227)
   (sqrt (* (* (* (fma -2.0 (/ (* l l) Om) t) n) U) 2.0))
   (* (sqrt t) (sqrt (* (* 2.0 n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.8e+227) {
		tmp = sqrt((((fma(-2.0, ((l * l) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(t) * sqrt(((2.0 * n) * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2.8e+227)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(2.0 * n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2.8e+227], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.79999999999999984e227

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.4

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

   3. Applied rewrites 53.6%

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

   5. Taylor expanded in n around 0

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

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

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

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 49.6%

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

   if 2.79999999999999984e227 < t

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.3

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 49.0

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 49.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.0

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

   6. Applied rewrites 49.0%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 59.8%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. unpow-prod-down N/A

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

       10. lower-*.f64 N/A

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

   11. Applied rewrites 84.8%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -4.1e-297)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* t (* 2.0 n))) (sqrt U))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -4.1e-297) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((t * (2.0 * n))) * sqrt(U);
	}
	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 (u <= (-4.1d-297)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((t * (2.0d0 * n))) * sqrt(u)
    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 (U <= -4.1e-297) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((t * (2.0 * n))) * Math.sqrt(U);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -4.1e-297:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((t * (2.0 * n))) * math.sqrt(U)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -4.1e-297)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(t * Float64(2.0 * n))) * sqrt(U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -4.1e-297)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((t * (2.0 * n))) * sqrt(U);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -4.1e-297], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -4.1000000000000002e-297

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.6

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

   3. Applied rewrites 53.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

   if -4.1000000000000002e-297 < U

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.5

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

   3. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 52.8

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 52.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 40.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. sqrt-prod N/A

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

       7. pow1/2 N/A

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

       8. lower-*.f64 N/A

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

   11. Applied rewrites 51.4%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{t \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -4.1e-297)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* 2.0 U)) (sqrt (* t n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -4.1e-297) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((t * 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 (u <= (-4.1d-297)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((t * 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 (U <= -4.1e-297) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((t * n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -4.1e-297:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((t * n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -4.1e-297)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(t * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -4.1e-297)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((2.0 * U)) * sqrt((t * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -4.1e-297], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -4.1000000000000002e-297

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.6

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

   3. Applied rewrites 53.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

   if -4.1000000000000002e-297 < U

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.5

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

   3. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 52.8

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 52.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

   6. Applied rewrites 52.8%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 40.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. sqrt-prod N/A

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

       10. pow1/2 N/A

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

       11. lower-*.f64 N/A

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

   11. Applied rewrites 51.4%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{t \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot 2} \cdot \sqrt{n \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1e-311)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* t 2.0)) (sqrt (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1e-311) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((t * 2.0)) * sqrt((n * U));
	}
	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 (t <= 1d-311) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((t * 2.0d0)) * sqrt((n * u))
    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 (t <= 1e-311) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((t * 2.0)) * Math.sqrt((n * U));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1e-311:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((t * 2.0)) * math.sqrt((n * U))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1e-311)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(t * 2.0)) * sqrt(Float64(n * U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1e-311)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((t * 2.0)) * sqrt((n * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1e-311], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999948e-312

   1. Initial program 52.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.2

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

   3. Applied rewrites 49.9%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 41.5

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

   7. Applied rewrites 41.5%

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

   if 9.99999999999948e-312 < t

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

   3. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 53.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 53.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.4

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 37.4%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. sqrt-prod N/A

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

   11. Applied rewrites 49.4%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot 2} \cdot \sqrt{n \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 53.5

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

3. Applied rewrites 53.6%

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

5. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 40.9

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

7. Applied rewrites 40.9%

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

8. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2024272 
(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*))))))