Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 63.1%

Time: 20.6s

Alternatives: 16

Speedup: 1.9×

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: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* - U, \frac{n \cdot \ell}{Om}, \ell \cdot -2\right), t\right) \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(U, n \cdot t, \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\ell, \frac{n \cdot \left(U* - U\right)}{Om}, \ell \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -9.6e-122)
   (sqrt
    (*
     n
     (* U (* (fma (/ l Om) (fma (- U* U) (/ (* n l) Om) (* l -2.0)) t) 2.0))))
   (if (<= n 5.5e-234)
     (sqrt
      (*
       2.0
       (fma
        U
        (* n t)
        (/ (* U (* (* n l) (fma l (/ (* n (- U* U)) Om) (* l -2.0)))) Om))))
     (*
      (sqrt (* n 2.0))
      (sqrt (* U (fma l (/ (* l (fma n (/ (- U* U) Om) -2.0)) Om) t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -9.6e-122) {
		tmp = sqrt((n * (U * (fma((l / Om), fma((U_42_ - U), ((n * l) / Om), (l * -2.0)), t) * 2.0))));
	} else if (n <= 5.5e-234) {
		tmp = sqrt((2.0 * fma(U, (n * t), ((U * ((n * l) * fma(l, ((n * (U_42_ - U)) / Om), (l * -2.0)))) / Om))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * fma(l, ((l * fma(n, ((U_42_ - U) / Om), -2.0)) / Om), t)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -9.6e-122)
		tmp = sqrt(Float64(n * Float64(U * Float64(fma(Float64(l / Om), fma(Float64(U_42_ - U), Float64(Float64(n * l) / Om), Float64(l * -2.0)), t) * 2.0))));
	elseif (n <= 5.5e-234)
		tmp = sqrt(Float64(2.0 * fma(U, Float64(n * t), Float64(Float64(U * Float64(Float64(n * l) * fma(l, Float64(Float64(n * Float64(U_42_ - U)) / Om), Float64(l * -2.0)))) / Om))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * fma(l, Float64(Float64(l * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)) / Om), t))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -9.6e-122], N[Sqrt[N[(n * N[(U * N[(N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 5.5e-234], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision] + N[(N[(U * N[(N[(n * l), $MachinePrecision] * N[(l * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(l * N[(N[(l * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.59999999999999949e-122

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 64.6%

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

   6. Applied egg-rr 65.4%

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

   if -9.59999999999999949e-122 < n < 5.5e-234

   1. Initial program 36.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 42.8%

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in t around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

   9. Simplified 66.3%

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

   if 5.5e-234 < n

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

   6. Applied egg-rr 62.2%

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

   8. Applied egg-rr 75.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* - U, \frac{n \cdot \ell}{Om}, \ell \cdot -2\right), t\right) \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(U, n \cdot t, \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\ell, \frac{n \cdot \left(U* - U\right)}{Om}, \ell \cdot -2\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(l * N[(N[(l * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(l * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 13.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 13.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

   6. Applied egg-rr 31.6%

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

   8. Applied egg-rr 52.1%

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

   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 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 75.0%

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

   6. Applied egg-rr 74.3%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 16.3%

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

   6. Applied egg-rr 27.7%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Simplified 47.7%

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

4. Final simplification 67.9%

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

Alternative 3: 63.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(l * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 13.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 24.3

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

   5. Simplified 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

   7. Applied egg-rr 24.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-prod N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-*.f64 N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 43.7

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

   9. Applied egg-rr 43.7%

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

   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 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 75.0%

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

   6. Applied egg-rr 74.3%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 16.3%

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

   6. Applied egg-rr 27.7%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Simplified 47.7%

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

4. Final simplification 67.0%

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

Alternative 4: 60.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(l / Om), $MachinePrecision] * N[(l * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(l * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 13.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 24.3

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

   5. Simplified 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

   7. Applied egg-rr 24.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-prod N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-*.f64 N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 43.7

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

   9. Applied egg-rr 43.7%

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

   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 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 75.0%

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

   6. Applied egg-rr 74.3%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 69.3

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

   9. Simplified 69.3%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 16.3%

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

   6. Applied egg-rr 27.7%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Simplified 47.7%

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

4. Final simplification 63.3%

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

Alternative 5: 60.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(l / Om), $MachinePrecision] * N[(l * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * N[(N[(n * l), $MachinePrecision] * N[(l * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 13.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 24.3

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

   5. Simplified 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

   7. Applied egg-rr 24.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-prod N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-*.f64 N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 43.7

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

   9. Applied egg-rr 43.7%

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

   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 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 75.0%

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

   6. Applied egg-rr 74.3%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 69.3

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

   9. Simplified 69.3%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 16.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

   6. Applied egg-rr 35.6%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Simplified 47.7%

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

4. Final simplification 63.3%

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

Alternative 6: 51.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \sqrt{\left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \cdot t\_2}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ l Om) (* l -2.0) t))
        (t_2 (* U (* n 2.0)))
        (t_3
         (sqrt
          (*
           (-
            (* (- U* U) (* n (pow (/ l Om) 2.0)))
            (- (* 2.0 (/ (* l l) Om)) t))
           t_2))))
   (if (<= t_3 0.0)
     (* (sqrt (* n 2.0)) (sqrt (* U t)))
     (if (<= t_3 4e+133) (sqrt (* t_2 t_1)) (sqrt (* U (* (* n 2.0) t_1)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((l / Om), (l * -2.0), t);
	double t_2 = U * (n * 2.0);
	double t_3 = sqrt(((((U_42_ - U) * (n * pow((l / Om), 2.0))) - ((2.0 * ((l * l) / Om)) - t)) * t_2));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	} else if (t_3 <= 4e+133) {
		tmp = sqrt((t_2 * t_1));
	} else {
		tmp = sqrt((U * ((n * 2.0) * t_1)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(l / Om), Float64(l * -2.0), t)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = sqrt(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * (Float64(l / Om) ^ 2.0))) - Float64(Float64(2.0 * Float64(Float64(l * l) / Om)) - t)) * t_2))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t)));
	elseif (t_3 <= 4e+133)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * t_1)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+133], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * t$95$1), $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 13.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 24.3

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

   5. Simplified 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

   7. Applied egg-rr 24.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-prod N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-*.f64 N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 43.7

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

   9. Applied egg-rr 43.7%

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

   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*))))) < 4.0000000000000001e133

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 98.8%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in n around 0

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

      1. lower-*.f64 93.9

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

   9. Simplified 93.9%

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

   if 4.0000000000000001e133 < (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 20.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in n around 0

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

      1. lower-*.f64 34.1

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

   9. Simplified 34.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 68.2%

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

   6. Applied egg-rr 64.9%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 1.4%

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

   6. Applied egg-rr 15.2%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Simplified 45.8%

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

4. Final simplification 62.4%

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

Alternative 8: 45.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right), t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.62e-170)
   (* (sqrt 2.0) (sqrt (* U (* n t))))
   (sqrt
    (* (* U (* n 2.0)) (fma (/ l Om) (* l (fma n (/ (- U* U) Om) -2.0)) t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.62e-170) {
		tmp = sqrt(2.0) * sqrt((U * (n * t)));
	} else {
		tmp = sqrt(((U * (n * 2.0)) * fma((l / Om), (l * fma(n, ((U_42_ - U) / Om), -2.0)), t)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.62e-170)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * fma(Float64(l / Om), Float64(l * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)), t)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.62e-170], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.62e-170

   1. Initial program 52.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 42.8

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

   5. Simplified 42.8%

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

   if 1.62e-170 < l

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 60.4%

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

   6. Applied egg-rr 60.7%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 60.9

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

   9. Simplified 60.9%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right), t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, \frac{\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.62e-170)
   (* (sqrt 2.0) (sqrt (* U (* n t))))
   (sqrt
    (* (fma l (/ (* l (fma n (/ (- U* U) Om) -2.0)) Om) t) (* U (* n 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.62e-170) {
		tmp = sqrt(2.0) * sqrt((U * (n * t)));
	} else {
		tmp = sqrt((fma(l, ((l * fma(n, ((U_42_ - U) / Om), -2.0)) / Om), t) * (U * (n * 2.0))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.62e-170)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(fma(l, Float64(Float64(l * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)) / Om), t) * Float64(U * Float64(n * 2.0))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.62e-170], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(l * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.62e-170

   1. Initial program 52.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 42.8

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

   5. Simplified 42.8%

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

   if 1.62e-170 < l

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

   6. Applied egg-rr 58.2%

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

   8. Applied egg-rr 60.9%

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

Alternative 10: 42.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.9e-23)
   (sqrt (* 2.0 (* t (* n U))))
   (sqrt (* U (* (* n 2.0) (fma (/ l Om) (* l -2.0) t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.9e-23) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt((U * ((n * 2.0) * fma((l / Om), (l * -2.0), t))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.9e-23)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(U * Float64(Float64(n * 2.0) * fma(Float64(l / Om), Float64(l * -2.0), t))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.9e-23], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.90000000000000006e-23

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 62.5%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 47.9

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

   7. Simplified 47.9%

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

   if 1.90000000000000006e-23 < l

   1. Initial program 39.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in n around 0

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

      1. lower-*.f64 49.2

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

   9. Simplified 49.2%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n \cdot 2\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot 2} \cdot \sqrt{n \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 5e+64)
   (sqrt (* 2.0 (* U (* n (fma -2.0 (/ (* l l) Om) t)))))
   (* (sqrt (* t 2.0)) (sqrt (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5e+64) {
		tmp = sqrt((2.0 * (U * (n * fma(-2.0, ((l * l) / Om), t)))));
	} else {
		tmp = sqrt((t * 2.0)) * sqrt((n * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 5e+64)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(-2.0, Float64(Float64(l * l) / Om), t)))));
	else
		tmp = Float64(sqrt(Float64(t * 2.0)) * sqrt(Float64(n * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 5e+64], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5e64

   1. Initial program 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-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. Applied egg-rr 57.7%

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

   6. Applied egg-rr 59.6%

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

   7. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.3

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

   9. Simplified 47.3%

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

   if 5e64 < t

   1. Initial program 57.5%

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

   4. Applied egg-rr 58.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 65.6

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

   7. Simplified 65.6%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot 2} \cdot \sqrt{n \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot 2} \cdot \sqrt{n \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.5e-254)
   (sqrt (* n (* t (* U 2.0))))
   (* (sqrt (* t 2.0)) (sqrt (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.5e-254) {
		tmp = sqrt((n * (t * (U * 2.0))));
	} else {
		tmp = sqrt((t * 2.0)) * sqrt((n * U));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.5d-254) then
        tmp = sqrt((n * (t * (u * 2.0d0))))
    else
        tmp = sqrt((t * 2.0d0)) * sqrt((n * u))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.5e-254) {
		tmp = Math.sqrt((n * (t * (U * 2.0))));
	} else {
		tmp = Math.sqrt((t * 2.0)) * Math.sqrt((n * U));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.5e-254:
		tmp = math.sqrt((n * (t * (U * 2.0))))
	else:
		tmp = math.sqrt((t * 2.0)) * math.sqrt((n * U))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.5e-254)
		tmp = sqrt(Float64(n * Float64(t * Float64(U * 2.0))));
	else
		tmp = Float64(sqrt(Float64(t * 2.0)) * sqrt(Float64(n * U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.5e-254)
		tmp = sqrt((n * (t * (U * 2.0))));
	else
		tmp = sqrt((t * 2.0)) * sqrt((n * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.5e-254], N[Sqrt[N[(n * N[(t * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.50000000000000006e-254

   1. Initial program 52.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 35.5

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

   5. Simplified 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

   7. Applied egg-rr 35.6%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 40.1

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

   9. Applied egg-rr 40.1%

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

   if 1.50000000000000006e-254 < t

   1. Initial program 50.8%

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

   4. Applied egg-rr 49.2%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 48.3

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

   7. Simplified 48.3%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot 2} \cdot \sqrt{n \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 38.6

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

5. Simplified 38.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-*r* N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

7. Applied egg-rr 38.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-unprod N/A

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

   13. lift-sqrt.f64 N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 39.0

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

9. Applied egg-rr 39.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot t\right)}\right) \cdot n} \]

    2. lift-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)} \cdot n} \]

    3. *-commutative N/A

       \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}} \]

    4. lift-*.f64 N/A

       \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)}} \]

    5. lift-*.f64 N/A

       \[\leadsto \sqrt{n \cdot \left(2 \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

    6. associate-*r* N/A

       \[\leadsto \sqrt{n \cdot \color{blue}{\left(\left(2 \cdot U\right) \cdot t\right)}} \]

    7. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot t}} \]

    8. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot 2\right) \cdot U\right)} \cdot t} \]

    9. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot 2\right)} \cdot U\right) \cdot t} \]

    10. *-commutative N/A

        \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right)} \cdot t} \]

    11. lift-*.f64 N/A

        \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right)} \cdot t} \]

    12. lower-*.f64 40.7

        \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t}} \]

11. Applied egg-rr 40.7%

    \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t}} \]

12. Final simplification 40.7%

    \[\leadsto \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \]
13. Add Preprocessing

Alternative 14: 35.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* t (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (t * (n * U))));
}

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 * (t * (n * u))))
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 * (t * (n * U))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (t * (n * U))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(t * Float64(n * U))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (t * (n * U))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. lift--.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   6. lift-pow.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

   8. lift--.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   10. sub-neg N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   11. +-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. Applied egg-rr 59.6%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

5. Taylor expanded in l around 0

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   4. lower-*.f64 40.7

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)} \]

7. Simplified 40.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]

8. Final simplification 40.7%

   \[\leadsto \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \]
9. Add Preprocessing

Alternative 15: 36.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 t around inf

   \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{U \cdot \left(n \cdot t\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot t\right)}} \]

   6. lower-*.f64 38.6

      \[\leadsto \sqrt{2} \cdot \sqrt{U \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Simplified 38.6%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{U \cdot \color{blue}{\left(n \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot t\right)}} \]

   3. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

   11. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   12. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

   15. lift-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   16. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

   17. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

   18. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

   19. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

   20. lift-*.f64 N/A

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   21. *-commutative N/A

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)} \]

   22. associate-*r* N/A

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   23. lift-*.f64 N/A

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   24. lift-*.f64 N/A

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
8. Add Preprocessing

Alternative 16: 5.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \left(U \cdot 2\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* n (* U 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((n * (U * 2.0)));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((n * (u * 2.0d0)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((n * (U * 2.0)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((n * (U * 2.0)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(n * Float64(U * 2.0)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((n * (U * 2.0)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(n * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 t around inf

   \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{U \cdot \left(n \cdot t\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot t\right)}} \]

   6. lower-*.f64 38.6

      \[\leadsto \sqrt{2} \cdot \sqrt{U \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Simplified 38.6%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot t\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{U \cdot \color{blue}{\left(n \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot t\right)}} \]

   3. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

   11. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   12. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

   15. lift-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   16. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

   17. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

   18. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

   19. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

   20. lift-*.f64 N/A

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   21. *-commutative N/A

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)} \]

   22. associate-*r* N/A

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   23. lift-*.f64 N/A

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   24. lift-*.f64 N/A

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(t \cdot n\right)}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot t\right) \cdot n\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot t\right)} \cdot n\right)} \]

   4. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\left(U \cdot t\right) \cdot n}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\left(U \cdot t\right) \cdot n} \]

   6. sqrt-unprod N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{n}\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{U \cdot t} \cdot \color{blue}{\sqrt{n}}\right) \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{U \cdot t}\right) \cdot \sqrt{n}} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sqrt{U \cdot t}\right) \cdot \sqrt{n} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\sqrt{U \cdot t}}\right) \cdot \sqrt{n} \]

   12. sqrt-unprod N/A

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot t\right)}} \cdot \sqrt{n} \]

   13. lift-sqrt.f64 N/A

       \[\leadsto \sqrt{2 \cdot \left(U \cdot t\right)} \cdot \color{blue}{\sqrt{n}} \]

   14. sqrt-unprod N/A

       \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n}} \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n}} \]

   16. lower-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n}} \]

   17. lower-*.f64 39.0

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot t\right)\right)} \cdot n} \]

9. Applied egg-rr 39.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n}} \]

10. Taylor expanded in U around inf

    \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot n} \]
11. Step-by-step derivation

    1. lower-*.f64 5.5

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot n} \]

12. Simplified 5.5%

    \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot n} \]

13. Final simplification 5.5%

    \[\leadsto \sqrt{n \cdot \left(U \cdot 2\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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*))))))