Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 61.2%

Time: 10.1s

Alternatives: 16

Speedup: 2.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{l\_m}{Om}, \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;l\_m \leq 3.2 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot l\_m\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot l\_m\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(n \cdot n\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, -2, -4 \cdot \frac{U \cdot n}{Om}\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.5e+68)
   (sqrt
    (*
     (fma
      (* (* (/ l_m Om) n) (- U* U))
      (/ l_m Om)
      (fma -2.0 (/ (* l_m l_m) Om) t))
     (* U (* n 2.0))))
   (if (<= l_m 3.2e+251)
     (sqrt
      (*
       (* (fma (/ (- U U*) Om) (/ n Om) (/ 2.0 Om)) l_m)
       (* (* (* -2.0 U) l_m) n)))
     (*
      (sqrt
       (fma
        (/ (* (* (* n n) U) (- U U*)) (* Om Om))
        -2.0
        (* -4.0 (/ (* U n) Om))))
      l_m))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e+68) {
		tmp = sqrt((fma((((l_m / Om) * n) * (U_42_ - U)), (l_m / Om), fma(-2.0, ((l_m * l_m) / Om), t)) * (U * (n * 2.0))));
	} else if (l_m <= 3.2e+251) {
		tmp = sqrt(((fma(((U - U_42_) / Om), (n / Om), (2.0 / Om)) * l_m) * (((-2.0 * U) * l_m) * n)));
	} else {
		tmp = sqrt(fma(((((n * n) * U) * (U - U_42_)) / (Om * Om)), -2.0, (-4.0 * ((U * n) / Om)))) * l_m;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.5e+68)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l_m / Om) * n) * Float64(U_42_ - U)), Float64(l_m / Om), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)) * Float64(U * Float64(n * 2.0))));
	elseif (l_m <= 3.2e+251)
		tmp = sqrt(Float64(Float64(fma(Float64(Float64(U - U_42_) / Om), Float64(n / Om), Float64(2.0 / Om)) * l_m) * Float64(Float64(Float64(-2.0 * U) * l_m) * n)));
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(n * n) * U) * Float64(U - U_42_)) / Float64(Om * Om)), -2.0, Float64(-4.0 * Float64(Float64(U * n) / Om)))) * l_m);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5.5e+68], N[Sqrt[N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.2e+251], N[Sqrt[N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(N[(N[(-2.0 * U), $MachinePrecision] * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(n * n), $MachinePrecision] * U), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.5000000000000004e68

   1. Initial program 54.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 56.4

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 56.4%

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

   if 5.5000000000000004e68 < l < 3.1999999999999997e251

   1. Initial program 24.0%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 34.2

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

   5. Applied rewrites 34.2%

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

   7. Applied rewrites 27.5%

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

   9. Applied rewrites 65.2%

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

   11. Applied rewrites 70.6%

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

   if 3.1999999999999997e251 < l

   1. Initial program 12.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 13.3

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 13.0

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

   6. Applied rewrites 13.0%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 77.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot \ell\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(n \cdot n\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, -2, -4 \cdot \frac{U \cdot n}{Om}\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2 \cdot \frac{l\_m}{Om}, l\_m, t\right) \cdot n\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \sqrt{\left(\left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_2}\\ t_4 := \frac{l\_m}{Om} \cdot n\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(l\_m \cdot l\_m\right)}{Om}\right) \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U* \cdot U\right) \cdot \left(t\_4 \cdot t\_4\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* -2.0 (/ l_m Om)) l_m t) n))
        (t_2 (* U (* n 2.0)))
        (t_3
         (sqrt
          (*
           (-
            (* (* (pow (/ l_m Om) 2.0) n) (- U* U))
            (- (* (/ (* l_m l_m) Om) 2.0) t))
           t_2)))
        (t_4 (* (/ l_m Om) n)))
   (if (<= t_3 0.0)
     (* (sqrt U) (sqrt (* t_1 2.0)))
     (if (<= t_3 5e+105)
       (sqrt (* (- t (/ (* (fma (- U U*) (/ n Om) 2.0) (* l_m l_m)) Om)) t_2))
       (if (<= t_3 INFINITY)
         (sqrt (* (* t_1 U) 2.0))
         (sqrt (* (* (* U* U) (* t_4 t_4)) 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((-2.0 * (l_m / Om)), l_m, t) * n;
	double t_2 = U * (n * 2.0);
	double t_3 = sqrt(((((pow((l_m / Om), 2.0) * n) * (U_42_ - U)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_2));
	double t_4 = (l_m / Om) * n;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(U) * sqrt((t_1 * 2.0));
	} else if (t_3 <= 5e+105) {
		tmp = sqrt(((t - ((fma((U - U_42_), (n / Om), 2.0) * (l_m * l_m)) / Om)) * t_2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t_1 * U) * 2.0));
	} else {
		tmp = sqrt((((U_42_ * U) * (t_4 * t_4)) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t) * n)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = sqrt(Float64(Float64(Float64(Float64((Float64(l_m / Om) ^ 2.0) * n) * Float64(U_42_ - U)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_2))
	t_4 = Float64(Float64(l_m / Om) * n)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * 2.0)));
	elseif (t_3 <= 5e+105)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(l_m * l_m)) / Om)) * t_2));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(t_1 * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(U_42_ * U) * Float64(t_4 * t_4)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * l$95$m + t), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+105], N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U$42$ * U), $MachinePrecision] * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 29.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000046e105

   1. Initial program 97.5%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 92.1%

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

   if 5.00000000000000046e105 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 35.4%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 33.7

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

   5. Applied rewrites 33.7%

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

   7. Applied rewrites 44.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 41.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U* \cdot U\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.4% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (-
            (* (* (pow (/ l_m Om) 2.0) n) (- U* U))
            (- (* (/ (* l_m l_m) Om) 2.0) t))
           (* U (* n 2.0)))))
        (t_2 (fma (* -2.0 (/ l_m Om)) l_m t))
        (t_3 (* (/ l_m Om) n)))
   (if (<= t_1 0.0)
     (* (sqrt U) (sqrt (* (* t_2 n) 2.0)))
     (if (<= t_1 INFINITY)
       (* (sqrt (* t_2 (* U n))) (sqrt 2.0))
       (sqrt (* (* (* U* U) (* t_3 t_3)) 2.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 29.1%

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

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

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 64.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 41.8%

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

4. Final simplification 56.2%

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

Alternative 4: 54.4% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 (/ l_m Om)) l_m t))
        (t_2
         (sqrt
          (*
           (-
            (* (* (pow (/ l_m Om) 2.0) n) (- U* U))
            (- (* (/ (* l_m l_m) Om) 2.0) t))
           (* U (* n 2.0))))))
   (if (<= t_2 0.0)
     (* (sqrt U) (sqrt (* (* t_1 n) 2.0)))
     (if (<= t_2 INFINITY)
       (* (sqrt (* t_1 (* U n))) (sqrt 2.0))
       (sqrt
        (* (* (* (/ n (* Om Om)) l_m) (- U*)) (* (* (* -2.0 U) n) l_m)))))))

C

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t)
	t_2 = sqrt(Float64(Float64(Float64(Float64((Float64(l_m / Om) ^ 2.0) * n) * Float64(U_42_ - U)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * Float64(U * Float64(n * 2.0))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(U) * sqrt(Float64(Float64(t_1 * n) * 2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(U * n))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(n / Float64(Om * Om)) * l_m) * Float64(-U_42_)) * Float64(Float64(Float64(-2.0 * U) * n) * l_m)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 29.1%

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

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

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 64.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 36.9%

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

   9. Applied rewrites 46.6%

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

   10. Taylor expanded in U* around inf

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

   12. Applied rewrites 36.6%

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

4. Final simplification 55.3%

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

Alternative 5: 54.7% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 (/ l_m Om)) l_m t))
        (t_2
         (sqrt
          (*
           (-
            (* (* (pow (/ l_m Om) 2.0) n) (- U* U))
            (- (* (/ (* l_m l_m) Om) 2.0) t))
           (* U (* n 2.0))))))
   (if (<= t_2 0.0)
     (* (sqrt U) (sqrt (* (* t_1 n) 2.0)))
     (if (<= t_2 INFINITY)
       (* (sqrt (* t_1 (* U n))) (sqrt 2.0))
       (sqrt (* (/ (* (* (* n l_m) (* n l_m)) (* U* U)) (* Om Om)) 2.0))))))

C

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t)
	t_2 = sqrt(Float64(Float64(Float64(Float64((Float64(l_m / Om) ^ 2.0) * n) * Float64(U_42_ - U)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * Float64(U * Float64(n * 2.0))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(U) * sqrt(Float64(Float64(t_1 * n) * 2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(U * n))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * l_m) * Float64(n * l_m)) * Float64(U_42_ * U)) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 29.1%

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

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

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 64.4%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 12.7

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 12.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.1

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

   7. Applied rewrites 34.1%

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

4. Final simplification 54.9%

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

Alternative 6: 53.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2 \cdot \frac{l\_m}{Om}, l\_m, t\right)\\ t_2 := \sqrt{\left(\left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{if}\;t\_2 \leq 10^{-83}:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot U\right) \cdot n} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(U \cdot n\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot l\_m\right) \cdot \left(n \cdot l\_m\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 (/ l_m Om)) l_m t))
        (t_2
         (sqrt
          (*
           (-
            (* (* (pow (/ l_m Om) 2.0) n) (- U* U))
            (- (* (/ (* l_m l_m) Om) 2.0) t))
           (* U (* n 2.0))))))
   (if (<= t_2 1e-83)
     (* (sqrt (* (* t_1 U) n)) (sqrt 2.0))
     (if (<= t_2 INFINITY)
       (* (sqrt (* t_1 (* U n))) (sqrt 2.0))
       (sqrt (* (/ (* (* (* n l_m) (* n l_m)) (* U* U)) (* Om Om)) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((-2.0 * (l_m / Om)), l_m, t);
	double t_2 = sqrt(((((pow((l_m / Om), 2.0) * n) * (U_42_ - U)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0))));
	double tmp;
	if (t_2 <= 1e-83) {
		tmp = sqrt(((t_1 * U) * n)) * sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (U * n))) * sqrt(2.0);
	} else {
		tmp = sqrt((((((n * l_m) * (n * l_m)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t)
	t_2 = sqrt(Float64(Float64(Float64(Float64((Float64(l_m / Om) ^ 2.0) * n) * Float64(U_42_ - U)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * Float64(U * Float64(n * 2.0))))
	tmp = 0.0
	if (t_2 <= 1e-83)
		tmp = Float64(sqrt(Float64(Float64(t_1 * U) * n)) * sqrt(2.0));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(U * n))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * l_m) * Float64(n * l_m)) * Float64(U_42_ * U)) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * l$95$m + t), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e-83], N[(N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e-83

   1. Initial program 37.4%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 35.0

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

   5. Applied rewrites 35.0%

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

   7. Applied rewrites 39.8%

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

   if 1e-83 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.9%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 63.6%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 12.7

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 12.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.1

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

   7. Applied rewrites 34.1%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{-83}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot U\right) \cdot n} \cdot \sqrt{2}\\ \mathbf{elif}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot \left(U \cdot n\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \sqrt{\left(\left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(t\_2 \cdot 2 - t\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_3 \leq 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{l\_m}{Om}, l\_m, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot l\_m\right) \cdot \left(n \cdot l\_m\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (sqrt
          (*
           (- (* (* (pow (/ l_m Om) 2.0) n) (- U* U)) (- (* t_2 2.0) t))
           t_1))))
   (if (<= t_3 1e+153)
     (sqrt (* (fma -2.0 t_2 t) t_1))
     (if (<= t_3 INFINITY)
       (sqrt (* (* (* (fma (* -2.0 (/ l_m Om)) l_m t) n) U) 2.0))
       (sqrt (* (/ (* (* (* n l_m) (* n l_m)) (* U* U)) (* Om Om)) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = sqrt(((((pow((l_m / Om), 2.0) * n) * (U_42_ - U)) - ((t_2 * 2.0) - t)) * t_1));
	double tmp;
	if (t_3 <= 1e+153) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((fma((-2.0 * (l_m / Om)), l_m, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((n * l_m) * (n * l_m)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = sqrt(Float64(Float64(Float64(Float64((Float64(l_m / Om) ^ 2.0) * n) * Float64(U_42_ - U)) - Float64(Float64(t_2 * 2.0) - t)) * t_1))
	tmp = 0.0
	if (t_3 <= 1e+153)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * l_m) * Float64(n * l_m)) * Float64(U_42_ * U)) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e+153], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * l$95$m + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e153

   1. Initial program 76.6%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 1e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 28.4%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 27.6

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

   5. Applied rewrites 27.6%

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

   7. Applied rewrites 39.9%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 12.7

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 12.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.1

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

   7. Applied rewrites 34.1%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\sqrt{\left(\left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2} \leq 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{l\_m}{Om}, l\_m, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om)) (t_2 (* U (* n 2.0))))
   (if (<=
        (sqrt
         (* (- (* (* (pow (/ l_m Om) 2.0) n) (- U* U)) (- (* t_1 2.0) t)) t_2))
        1e+153)
     (sqrt (* (fma -2.0 t_1 t) t_2))
     (sqrt (* (* (* (fma (* -2.0 (/ l_m Om)) l_m t) n) U) 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = U * (n * 2.0);
	double tmp;
	if (sqrt(((((pow((l_m / Om), 2.0) * n) * (U_42_ - U)) - ((t_1 * 2.0) - t)) * t_2)) <= 1e+153) {
		tmp = sqrt((fma(-2.0, t_1, t) * t_2));
	} else {
		tmp = sqrt((((fma((-2.0 * (l_m / Om)), l_m, t) * n) * U) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(Float64((Float64(l_m / Om) ^ 2.0) * n) * Float64(U_42_ - U)) - Float64(Float64(t_1 * 2.0) - t)) * t_2)) <= 1e+153)
		tmp = sqrt(Float64(fma(-2.0, t_1, t) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t) * n) * U) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], 1e+153], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * l$95$m + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e153

   1. Initial program 76.6%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 1e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 18.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 19.6

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

   5. Applied rewrites 19.6%

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

   7. Applied rewrites 31.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U* - U\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.4 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot n, U* - U, -2 \cdot l\_m\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{elif}\;l\_m \leq 3.2 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot l\_m\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot l\_m\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(n \cdot n\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, -2, -4 \cdot \frac{U \cdot n}{Om}\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8.4e+151)
   (sqrt
    (*
     (*
      (fma (/ l_m Om) (fma (* (/ l_m Om) n) (- U* U) (* -2.0 l_m)) t)
      (* n 2.0))
     U))
   (if (<= l_m 3.2e+251)
     (sqrt
      (*
       (* (fma (/ (- U U*) Om) (/ n Om) (/ 2.0 Om)) l_m)
       (* (* (* -2.0 U) l_m) n)))
     (*
      (sqrt
       (fma
        (/ (* (* (* n n) U) (- U U*)) (* Om Om))
        -2.0
        (* -4.0 (/ (* U n) Om))))
      l_m))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8.4e+151) {
		tmp = sqrt(((fma((l_m / Om), fma(((l_m / Om) * n), (U_42_ - U), (-2.0 * l_m)), t) * (n * 2.0)) * U));
	} else if (l_m <= 3.2e+251) {
		tmp = sqrt(((fma(((U - U_42_) / Om), (n / Om), (2.0 / Om)) * l_m) * (((-2.0 * U) * l_m) * n)));
	} else {
		tmp = sqrt(fma(((((n * n) * U) * (U - U_42_)) / (Om * Om)), -2.0, (-4.0 * ((U * n) / Om)))) * l_m;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8.4e+151)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(l_m / Om) * n), Float64(U_42_ - U), Float64(-2.0 * l_m)), t) * Float64(n * 2.0)) * U));
	elseif (l_m <= 3.2e+251)
		tmp = sqrt(Float64(Float64(fma(Float64(Float64(U - U_42_) / Om), Float64(n / Om), Float64(2.0 / Om)) * l_m) * Float64(Float64(Float64(-2.0 * U) * l_m) * n)));
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(n * n) * U) * Float64(U - U_42_)) / Float64(Om * Om)), -2.0, Float64(-4.0 * Float64(Float64(U * n) / Om)))) * l_m);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 8.4e+151], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.2e+251], N[Sqrt[N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(N[(N[(-2.0 * U), $MachinePrecision] * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(n * n), $MachinePrecision] * U), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.4000000000000002e151

   1. Initial program 54.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.7

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 55.7%

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

   6. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 61.1%

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

   if 8.4000000000000002e151 < l < 3.1999999999999997e251

   1. Initial program 12.4%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 26.7%

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

   9. Applied rewrites 67.0%

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

   11. Applied rewrites 70.3%

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

   if 3.1999999999999997e251 < l

   1. Initial program 12.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 13.3

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 13.0

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

   6. Applied rewrites 13.0%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 77.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.4 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U* - U, -2 \cdot \ell\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot \ell\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(n \cdot n\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, -2, -4 \cdot \frac{U \cdot n}{Om}\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot n, U* - U, -2 \cdot l\_m\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.3e+204)
   (sqrt
    (*
     (*
      (fma (/ l_m Om) (fma (* (/ l_m Om) n) (- U* U) (* -2.0 l_m)) t)
      (* n 2.0))
     U))
   (*
    (* (sqrt 2.0) l_m)
    (sqrt (* (* (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om)) n) U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.3e+204) {
		tmp = sqrt(((fma((l_m / Om), fma(((l_m / Om) * n), (U_42_ - U), (-2.0 * l_m)), t) * (n * 2.0)) * U));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om)) * n) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.3e+204)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(l_m / Om) * n), Float64(U_42_ - U), Float64(-2.0 * l_m)), t) * Float64(n * 2.0)) * U));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om)) * n) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.3e+204], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2999999999999999e204

   1. Initial program 51.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.3

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 55.5%

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

   6. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 61.4%

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

   if 2.2999999999999999e204 < l

   1. Initial program 14.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 26.9

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 26.9%

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 69.9%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U* - U, -2 \cdot \ell\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot n, U* - U, -2 \cdot l\_m\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \frac{l\_m}{Om}, l\_m, t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 1.05e+120)
   (sqrt
    (*
     (*
      (fma (/ l_m Om) (fma (* (/ l_m Om) n) (- U* U) (* -2.0 l_m)) t)
      (* n 2.0))
     U))
   (* (sqrt (* U n)) (sqrt (* (fma (* -2.0 (/ l_m Om)) l_m t) 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 1.05e+120) {
		tmp = sqrt(((fma((l_m / Om), fma(((l_m / Om) * n), (U_42_ - U), (-2.0 * l_m)), t) * (n * 2.0)) * U));
	} else {
		tmp = sqrt((U * n)) * sqrt((fma((-2.0 * (l_m / Om)), l_m, t) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 1.05e+120)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(l_m / Om) * n), Float64(U_42_ - U), Float64(-2.0 * l_m)), t) * Float64(n * 2.0)) * U));
	else
		tmp = Float64(sqrt(Float64(U * n)) * sqrt(Float64(fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t) * 2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 1.05e+120], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * l$95$m + t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e120

   1. Initial program 47.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 54.5

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 53.0%

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

   6. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 61.5%

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

   if 1.05e120 < 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. Add Preprocessing

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.2

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

   5. Applied rewrites 44.2%

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

   7. Applied rewrites 71.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U* - U, -2 \cdot \ell\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{l\_m}{Om}, l\_m, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{l\_m}{Om} \cdot 2\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot n\right) \cdot l\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.5e+203)
   (sqrt (* (* (* (fma (* -2.0 (/ l_m Om)) l_m t) n) U) 2.0))
   (sqrt (* (* (/ l_m Om) 2.0) (* (* (* -2.0 U) n) l_m)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.5e+203) {
		tmp = sqrt((((fma((-2.0 * (l_m / Om)), l_m, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((l_m / Om) * 2.0) * (((-2.0 * U) * n) * l_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.5e+203)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(-2.0 * Float64(l_m / Om)), l_m, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(l_m / Om) * 2.0) * Float64(Float64(Float64(-2.0 * U) * n) * l_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.5e+203], N[Sqrt[N[(N[(N[(N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * l$95$m + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l$95$m / Om), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(-2.0 * U), $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.50000000000000031e203

   1. Initial program 51.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.6

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

   5. Applied rewrites 44.6%

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

   7. Applied rewrites 49.8%

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

   if 3.50000000000000031e203 < l

   1. Initial program 14.5%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \color{blue}{\frac{n}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      19. lower-/.f64 36.8

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 36.8%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right) \cdot \color{blue}{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.1%

      \[\leadsto \sqrt{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)\right)}} \]

   10. Taylor expanded in n around 0

       \[\leadsto \sqrt{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\frac{\ell}{Om}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 40.0%

       \[\leadsto \sqrt{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell}{Om} \cdot 2\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot n\right) \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.7 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{l\_m}{Om} \cdot 2\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot n\right) \cdot l\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4.7e+36)
   (sqrt (* (* (* U n) t) 2.0))
   (sqrt (* (* (/ l_m Om) 2.0) (* (* (* -2.0 U) n) l_m)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.7e+36) {
		tmp = sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = sqrt((((l_m / Om) * 2.0) * (((-2.0 * U) * n) * l_m)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 4.7d+36) then
        tmp = sqrt((((u * n) * t) * 2.0d0))
    else
        tmp = sqrt((((l_m / om) * 2.0d0) * ((((-2.0d0) * u) * n) * l_m)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 4.7e+36) {
		tmp = Math.sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = Math.sqrt((((l_m / Om) * 2.0) * (((-2.0 * U) * n) * l_m)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 4.7e+36:
		tmp = math.sqrt((((U * n) * t) * 2.0))
	else:
		tmp = math.sqrt((((l_m / Om) * 2.0) * (((-2.0 * U) * n) * l_m)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4.7e+36)
		tmp = sqrt(Float64(Float64(Float64(U * n) * t) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(l_m / Om) * 2.0) * Float64(Float64(Float64(-2.0 * U) * n) * l_m)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 4.7e+36)
		tmp = sqrt((((U * n) * t) * 2.0));
	else
		tmp = sqrt((((l_m / Om) * 2.0) * (((-2.0 * U) * n) * l_m)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 4.7e+36], N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l$95$m / Om), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(-2.0 * U), $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.69999999999999989e36

   1. Initial program 54.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 39.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 39.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.7%

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \]

   if 4.69999999999999989e36 < l

   1. Initial program 22.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{\left(U - U*\right) \cdot n}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. times-frac N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{\frac{U - U*}{Om} \cdot \frac{n}{Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{U - U*}{Om}}, \frac{n}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{U - U*}}{Om}, \frac{n}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \color{blue}{\frac{n}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      19. lower-/.f64 35.2

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 35.2%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right) \cdot \color{blue}{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.4%

      \[\leadsto \sqrt{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)\right)}} \]

   10. Taylor expanded in n around 0

       \[\leadsto \sqrt{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\frac{\ell}{Om}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 40.8%

       \[\leadsto \sqrt{\left(\left(\left(U \cdot -2\right) \cdot n\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell}{Om} \cdot 2\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot n\right) \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(l\_m \cdot l\_m\right) \cdot n}{Om} \cdot U\right) \cdot -4}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.5e+68)
   (sqrt (* (* (* U n) t) 2.0))
   (sqrt (* (* (/ (* (* l_m l_m) n) Om) U) -4.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.5e+68) {
		tmp = sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.5d+68) then
        tmp = sqrt((((u * n) * t) * 2.0d0))
    else
        tmp = sqrt((((((l_m * l_m) * n) / om) * u) * (-4.0d0)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.5e+68) {
		tmp = Math.sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = Math.sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.5e+68:
		tmp = math.sqrt((((U * n) * t) * 2.0))
	else:
		tmp = math.sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.5e+68)
		tmp = sqrt(Float64(Float64(Float64(U * n) * t) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) / Om) * U) * -4.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.5e+68)
		tmp = sqrt((((U * n) * t) * 2.0));
	else
		tmp = sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.5e+68], N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * U), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5000000000000001e68

   1. Initial program 54.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 39.0

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 39.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.2%

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \]

   if 1.5000000000000001e68 < l

   1. Initial program 21.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{\left(U - U*\right) \cdot n}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. times-frac N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{\frac{U - U*}{Om} \cdot \frac{n}{Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{U - U*}{Om}}, \frac{n}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{U - U*}}{Om}, \frac{n}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \color{blue}{\frac{n}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      19. lower-/.f64 34.9

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 34.9%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.4%

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\left(U \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om} \cdot U\right) \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* U n) t) 2.0)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((((U * n) * t) * 2.0));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((u * n) * t) * 2.0d0))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((((U * n) * t) * 2.0));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((U * n) * t) * 2.0))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(U * n) * t) * 2.0))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((U * n) * t) * 2.0));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. *-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 33.9

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 33.9%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 35.7%

   \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \]
8. Add Preprocessing

Alternative 16: 35.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(U \cdot 2\right) \cdot t\right) \cdot n} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* U 2.0) t) n)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((((U * 2.0) * t) * n));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((u * 2.0d0) * t) * n))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((((U * 2.0) * t) * n));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((U * 2.0) * t) * n))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(U * 2.0) * t) * n))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((U * 2.0) * t) * n));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. *-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 33.9

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 33.9%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 32.5%

   \[\leadsto \sqrt{n \cdot \color{blue}{\left(t \cdot \left(2 \cdot U\right)\right)}} \]

8. Final simplification 32.5%

   \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot t\right) \cdot n} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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*))))))