Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 65.3%

Time: 23.1s

Alternatives: 25

Speedup: 1.7×

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: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if U < -9.999999999999969e-311

   1. Initial program 52.6%

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

   3. Simplified 61.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 62.5%

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

   if -9.999999999999969e-311 < U

   1. Initial program 45.3%

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

   3. Simplified 56.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot n\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{U \cdot 2}\right)}\right) \]

   6. Applied egg-rr 76.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(n \cdot \left(\frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}} + t\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(n \cdot \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}} + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{U}, 2\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}} \cdot n + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot n + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right) \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot n\right) + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right) \cdot \frac{1}{\frac{\frac{Om}{\ell}}{n}} + n \cdot t\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      9. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{fma}\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2, \frac{1}{\frac{\frac{Om}{\ell}}{n}}, n \cdot t\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{U}, 2\right)\right)\right) \]

      10. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\left(\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right), \left(\frac{1}{\frac{\frac{Om}{\ell}}{n}}\right), \left(n \cdot t\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{U}, 2\right)\right)\right) \]

   8. Applied egg-rr 80.3%

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

4. Final simplification 71.7%

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

Alternative 2: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \left(t + \frac{\ell \cdot -2 + \left(\ell \cdot n\right) \cdot \frac{U*}{Om}}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.32e-114)
   (sqrt
    (*
     n
     (*
      U
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (if (<= n 7.2e-240)
     (sqrt (+ (* (* t n) (* U 2.0)) (* -4.0 (/ (* (* l n) (* U l)) Om))))
     (*
      (sqrt n)
      (sqrt
       (*
        (* U 2.0)
        (+ t (/ (+ (* l -2.0) (* (* l n) (/ U* Om))) (/ Om l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.32e-114) {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else if (n <= 7.2e-240) {
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * (((l * n) * (U * l)) / Om))));
	} else {
		tmp = sqrt(n) * sqrt(((U * 2.0) * (t + (((l * -2.0) + ((l * n) * (U_42_ / Om))) / (Om / l)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-1.32d-114)) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else if (n <= 7.2d-240) then
        tmp = sqrt((((t * n) * (u * 2.0d0)) + ((-4.0d0) * (((l * n) * (u * l)) / om))))
    else
        tmp = sqrt(n) * sqrt(((u * 2.0d0) * (t + (((l * (-2.0d0)) + ((l * n) * (u_42 / om))) / (om / l)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.32e-114) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else if (n <= 7.2e-240) {
		tmp = Math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * (((l * n) * (U * l)) / Om))));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt(((U * 2.0) * (t + (((l * -2.0) + ((l * n) * (U_42_ / Om))) / (Om / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -1.32e-114:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	elif n <= 7.2e-240:
		tmp = math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * (((l * n) * (U * l)) / Om))))
	else:
		tmp = math.sqrt(n) * math.sqrt(((U * 2.0) * (t + (((l * -2.0) + ((l * n) * (U_42_ / Om))) / (Om / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.32e-114)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l)))))));
	elseif (n <= 7.2e-240)
		tmp = sqrt(Float64(Float64(Float64(t * n) * Float64(U * 2.0)) + Float64(-4.0 * Float64(Float64(Float64(l * n) * Float64(U * l)) / Om))));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(U * 2.0) * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(l * n) * Float64(U_42_ / Om))) / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -1.32e-114)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	elseif (n <= 7.2e-240)
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * (((l * n) * (U * l)) / Om))));
	else
		tmp = sqrt(n) * sqrt(((U * 2.0) * (t + (((l * -2.0) + ((l * n) * (U_42_ / Om))) / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.32e-114], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 7.2e-240], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(N[(l * n), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(l * n), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.31999999999999996e-114

   1. Initial program 63.1%

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

   3. Simplified 71.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 76.5%

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

   if -1.31999999999999996e-114 < n < 7.1999999999999998e-240

   1. Initial program 42.2%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 52.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 63.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 63.8%

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

   if 7.1999999999999998e-240 < n

   1. Initial program 42.2%

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

   3. Simplified 57.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot n\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{U \cdot 2}\right)}\right) \]

   6. Applied egg-rr 43.8%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      3. *-lowering-*.f64 42.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

   9. Simplified 42.0%

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

       1. pow1/2 N/A

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

       2. unpow-prod-down N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left({\left(t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}\right)}^{\frac{1}{2}} \cdot \sqrt{U \cdot 2}\right)}\right) \]

       5. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n}\right), \left(\color{blue}{{\left(t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}\right)}^{\frac{1}{2}}} \cdot \sqrt{U \cdot 2}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\color{blue}{{\left(t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}\right)}^{\frac{1}{2}}} \cdot \sqrt{U \cdot 2}\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}} \cdot \sqrt{\color{blue}{U \cdot 2}}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}} \cdot \sqrt{2 \cdot U}\right)\right) \]

       9. sqrt-unprod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\left(t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot U\right)}\right)\right) \]

       10. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot U\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + \ell \cdot -2}{\frac{Om}{\ell}}\right), \left(2 \cdot U\right)\right)\right)\right) \]

   11. Applied egg-rr 62.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \left(t + \frac{\ell \cdot -2 + \left(\ell \cdot n\right) \cdot \frac{U*}{Om}}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l))
    if (u <= (-1d-310)) then
        tmp = sqrt((n * (u * (2.0d0 * t_1))))
    else
        tmp = sqrt((u * 2.0d0)) * sqrt((n * t_1))
    end if
    code = tmp
end function

Java

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

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l))
	tmp = 0
	if U <= -1e-310:
		tmp = math.sqrt((n * (U * (2.0 * t_1))))
	else:
		tmp = math.sqrt((U * 2.0)) * math.sqrt((n * t_1))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l));
	tmp = 0.0;
	if (U <= -1e-310)
		tmp = sqrt((n * (U * (2.0 * t_1))));
	else
		tmp = sqrt((U * 2.0)) * sqrt((n * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if U < -9.999999999999969e-311

   1. Initial program 52.6%

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

   3. Simplified 61.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 62.5%

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

   if -9.999999999999969e-311 < U

   1. Initial program 45.3%

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

   3. Simplified 56.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot n\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{U \cdot 2}\right)}\right) \]

   6. Applied egg-rr 76.1%

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

4. Final simplification 69.5%

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

Alternative 4: 63.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-1d-310)) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt((u * 2.0d0)) * sqrt((n * (t + (((l * (-2.0d0)) + ((u_42 * l) * (n / om))) / (om / l)))))
    end if
    code = tmp
end function

Java

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

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -1e-310:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt((U * 2.0)) * math.sqrt((n * (t + (((l * -2.0) + ((U_42_ * l) * (n / Om))) / (Om / l)))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -1e-310)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt((U * 2.0)) * sqrt((n * (t + (((l * -2.0) + ((U_42_ * l) * (n / Om))) / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if U < -9.999999999999969e-311

   1. Initial program 52.6%

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

   3. Simplified 61.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 62.5%

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

   if -9.999999999999969e-311 < U

   1. Initial program 45.3%

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

   3. Simplified 56.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot n\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{U \cdot 2}\right)}\right) \]

   6. Applied egg-rr 76.1%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      3. *-lowering-*.f64 73.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

   9. Simplified 73.1%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(U* \cdot \ell\right) \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(U* \cdot \ell\right) \cdot \frac{n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(U* \cdot \ell\right), \left(\frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \ell\right), \left(\frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

       5. /-lowering-/.f64 73.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \ell\right), \mathsf{/.f64}\left(n, Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

   11. Applied egg-rr 73.9%

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

4. Final simplification 68.4%

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

Alternative 5: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 4.45 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 4.45e+105)
   (sqrt
    (*
     n
     (*
      U
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (* (sqrt (* U 2.0)) (sqrt (* n (+ t (/ (* -2.0 (* l l)) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 4.45e+105) {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = sqrt((U * 2.0)) * sqrt((n * (t + ((-2.0 * (l * l)) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 4.45d+105) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt((u * 2.0d0)) * sqrt((n * (t + (((-2.0d0) * (l * l)) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 4.45e+105) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = Math.sqrt((U * 2.0)) * Math.sqrt((n * (t + ((-2.0 * (l * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 4.45e+105:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt((U * 2.0)) * math.sqrt((n * (t + ((-2.0 * (l * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 4.45e+105)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l)))))));
	else
		tmp = Float64(sqrt(Float64(U * 2.0)) * sqrt(Float64(n * Float64(t + Float64(Float64(-2.0 * Float64(l * l)) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 4.45e+105)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt((U * 2.0)) * sqrt((n * (t + ((-2.0 * (l * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 4.45e+105], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * N[(t + N[(N[(-2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 4.44999999999999987e105

   1. Initial program 47.2%

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

   3. Simplified 57.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 62.5%

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

   if 4.44999999999999987e105 < U

   1. Initial program 57.3%

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

   3. Simplified 64.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot n\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{U \cdot 2}\right)}\right) \]

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

      8. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(U, 2\right)\right)\right) \]

   9. Simplified 71.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.45 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.55 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.55e-250)
   (sqrt
    (*
     n
     (*
      U
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (if (<= n 4.4e-167)
     (sqrt (+ (* (* t n) (* U 2.0)) (* -4.0 (* (* l (* l n)) (/ U Om)))))
     (sqrt
      (*
       2.0
       (* (+ t (* (/ l Om) (+ (* l -2.0) (* U* (/ (* l n) Om))))) (* U n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.55e-250) {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else if (n <= 4.4e-167) {
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	} else {
		tmp = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-2.55d-250)) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else if (n <= 4.4d-167) then
        tmp = sqrt((((t * n) * (u * 2.0d0)) + ((-4.0d0) * ((l * (l * n)) * (u / om)))))
    else
        tmp = sqrt((2.0d0 * ((t + ((l / om) * ((l * (-2.0d0)) + (u_42 * ((l * n) / om))))) * (u * n))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.55e-250) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else if (n <= 4.4e-167) {
		tmp = Math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -2.55e-250:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	elif n <= 4.4e-167:
		tmp = math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))))
	else:
		tmp = math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -2.55e-250)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l)))))));
	elseif (n <= 4.4e-167)
		tmp = sqrt(Float64(Float64(Float64(t * n) * Float64(U * 2.0)) + Float64(-4.0 * Float64(Float64(l * Float64(l * n)) * Float64(U / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(l * n) / Om))))) * Float64(U * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -2.55e-250)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	elseif (n <= 4.4e-167)
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	else
		tmp = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -2.55e-250], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.4e-167], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.5500000000000001e-250

   1. Initial program 59.5%

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

   3. Simplified 67.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 71.9%

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

   if -2.5500000000000001e-250 < n < 4.3999999999999999e-167

   1. Initial program 34.7%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 49.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 49.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(n \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 58.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(U, Om\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 58.5%

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

   if 4.3999999999999999e-167 < n

   1. Initial program 44.5%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 61.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.55 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left(t + \frac{t\_1}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot t\_1\right) \cdot \left(U \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ (* l -2.0) (* U* (/ (* l n) Om)))))
   (if (<= n -3e-206)
     (sqrt (* (+ t (/ t_1 (/ Om l))) (* n (* U 2.0))))
     (if (<= n 4e-167)
       (sqrt (+ (* (* t n) (* U 2.0)) (* -4.0 (* (* l (* l n)) (/ U Om)))))
       (sqrt (* 2.0 (* (+ t (* (/ l Om) t_1)) (* U n))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + (U_42_ * ((l * n) / Om));
	double tmp;
	if (n <= -3e-206) {
		tmp = sqrt(((t + (t_1 / (Om / l))) * (n * (U * 2.0))));
	} else if (n <= 4e-167) {
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	} else {
		tmp = sqrt((2.0 * ((t + ((l / Om) * t_1)) * (U * n))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * (-2.0d0)) + (u_42 * ((l * n) / om))
    if (n <= (-3d-206)) then
        tmp = sqrt(((t + (t_1 / (om / l))) * (n * (u * 2.0d0))))
    else if (n <= 4d-167) then
        tmp = sqrt((((t * n) * (u * 2.0d0)) + ((-4.0d0) * ((l * (l * n)) * (u / om)))))
    else
        tmp = sqrt((2.0d0 * ((t + ((l / om) * t_1)) * (u * n))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + (U_42_ * ((l * n) / Om));
	double tmp;
	if (n <= -3e-206) {
		tmp = Math.sqrt(((t + (t_1 / (Om / l))) * (n * (U * 2.0))));
	} else if (n <= 4e-167) {
		tmp = Math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * ((t + ((l / Om) * t_1)) * (U * n))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * -2.0) + (U_42_ * ((l * n) / Om))
	tmp = 0
	if n <= -3e-206:
		tmp = math.sqrt(((t + (t_1 / (Om / l))) * (n * (U * 2.0))))
	elif n <= 4e-167:
		tmp = math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))))
	else:
		tmp = math.sqrt((2.0 * ((t + ((l / Om) * t_1)) * (U * n))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(l * n) / Om)))
	tmp = 0.0
	if (n <= -3e-206)
		tmp = sqrt(Float64(Float64(t + Float64(t_1 / Float64(Om / l))) * Float64(n * Float64(U * 2.0))));
	elseif (n <= 4e-167)
		tmp = sqrt(Float64(Float64(Float64(t * n) * Float64(U * 2.0)) + Float64(-4.0 * Float64(Float64(l * Float64(l * n)) * Float64(U / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(t + Float64(Float64(l / Om) * t_1)) * Float64(U * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * -2.0) + (U_42_ * ((l * n) / Om));
	tmp = 0.0;
	if (n <= -3e-206)
		tmp = sqrt(((t + (t_1 / (Om / l))) * (n * (U * 2.0))));
	elseif (n <= 4e-167)
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	else
		tmp = sqrt((2.0 * ((t + ((l / Om) * t_1)) * (U * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e-206], N[Sqrt[N[(N[(t + N[(t$95$1 / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4e-167], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(t + N[(N[(l / Om), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.0000000000000002e-206

   1. Initial program 63.1%

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

   3. Simplified 70.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 73.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot 2\right) \cdot U\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot U\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot 2\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot \left(U \cdot 2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot n\right) \cdot \left(U \cdot 2\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right), \left(n \cdot \left(U \cdot 2\right)\right)\right)\right) \]

   8. Applied egg-rr 70.9%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       4. *-lowering-*.f64 71.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

   11. Simplified 71.0%

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

   if -3.0000000000000002e-206 < n < 4.00000000000000001e-167

   1. Initial program 32.9%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 47.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 47.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(n \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 57.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(U, Om\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 57.4%

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

   if 4.00000000000000001e-167 < n

   1. Initial program 44.5%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 61.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \mathbf{if}\;n \leq -3.1 \cdot 10^{-206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           2.0
           (*
            (+ t (* (/ l Om) (+ (* l -2.0) (* U* (/ (* l n) Om)))))
            (* U n))))))
   (if (<= n -3.1e-206)
     t_1
     (if (<= n 3.6e-167)
       (sqrt (+ (* (* t n) (* U 2.0)) (* -4.0 (* (* l (* l n)) (/ U Om)))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))));
	double tmp;
	if (n <= -3.1e-206) {
		tmp = t_1;
	} else if (n <= 3.6e-167) {
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * ((t + ((l / om) * ((l * (-2.0d0)) + (u_42 * ((l * n) / om))))) * (u * n))))
    if (n <= (-3.1d-206)) then
        tmp = t_1
    else if (n <= 3.6d-167) then
        tmp = sqrt((((t * n) * (u * 2.0d0)) + ((-4.0d0) * ((l * (l * n)) * (u / om)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))));
	double tmp;
	if (n <= -3.1e-206) {
		tmp = t_1;
	} else if (n <= 3.6e-167) {
		tmp = Math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))))
	tmp = 0
	if n <= -3.1e-206:
		tmp = t_1
	elif n <= 3.6e-167:
		tmp = math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(l * n) / Om))))) * Float64(U * n))))
	tmp = 0.0
	if (n <= -3.1e-206)
		tmp = t_1;
	elseif (n <= 3.6e-167)
		tmp = sqrt(Float64(Float64(Float64(t * n) * Float64(U * 2.0)) + Float64(-4.0 * Float64(Float64(l * Float64(l * n)) * Float64(U / Om)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((l * n) / Om))))) * (U * n))));
	tmp = 0.0;
	if (n <= -3.1e-206)
		tmp = t_1;
	elseif (n <= 3.6e-167)
		tmp = sqrt((((t * n) * (U * 2.0)) + (-4.0 * ((l * (l * n)) * (U / Om)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3.1e-206], t$95$1, If[LessEqual[n, 3.6e-167], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.1000000000000003e-206 or 3.6000000000000001e-167 < n

   1. Initial program 54.6%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 66.7%

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

   if -3.1000000000000003e-206 < n < 3.6000000000000001e-167

   1. Initial program 32.9%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 47.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 47.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(n \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 57.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(U, Om\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 57.4%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot n\right) \cdot \left(U \cdot 2\right)\\ \mathbf{if}\;Om \leq -2.7 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot -2 + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* t n) (* U 2.0))))
   (if (<= Om -2.7e-132)
     (sqrt (+ t_1 (* -4.0 (* l (* n (/ (* U l) Om))))))
     (if (<= Om 1.5e-100)
       (sqrt
        (/ (* 2.0 (* U (* (* l n) (+ (* l -2.0) (/ (* U* (* l n)) Om))))) Om))
       (sqrt (+ t_1 (* -4.0 (/ (* (* l n) (* U l)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -2.7e-132) {
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 1.5e-100) {
		tmp = sqrt(((2.0 * (U * ((l * n) * ((l * -2.0) + ((U_42_ * (l * n)) / Om))))) / Om));
	} else {
		tmp = sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * n) * (u * 2.0d0)
    if (om <= (-2.7d-132)) then
        tmp = sqrt((t_1 + ((-4.0d0) * (l * (n * ((u * l) / om))))))
    else if (om <= 1.5d-100) then
        tmp = sqrt(((2.0d0 * (u * ((l * n) * ((l * (-2.0d0)) + ((u_42 * (l * n)) / om))))) / om))
    else
        tmp = sqrt((t_1 + ((-4.0d0) * (((l * n) * (u * l)) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -2.7e-132) {
		tmp = Math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 1.5e-100) {
		tmp = Math.sqrt(((2.0 * (U * ((l * n) * ((l * -2.0) + ((U_42_ * (l * n)) / Om))))) / Om));
	} else {
		tmp = Math.sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (t * n) * (U * 2.0)
	tmp = 0
	if Om <= -2.7e-132:
		tmp = math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))))
	elif Om <= 1.5e-100:
		tmp = math.sqrt(((2.0 * (U * ((l * n) * ((l * -2.0) + ((U_42_ * (l * n)) / Om))))) / Om))
	else:
		tmp = math.sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(t * n) * Float64(U * 2.0))
	tmp = 0.0
	if (Om <= -2.7e-132)
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(l * Float64(n * Float64(Float64(U * l) / Om))))));
	elseif (Om <= 1.5e-100)
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(l * n) * Float64(Float64(l * -2.0) + Float64(Float64(U_42_ * Float64(l * n)) / Om))))) / Om));
	else
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(Float64(Float64(l * n) * Float64(U * l)) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (t * n) * (U * 2.0);
	tmp = 0.0;
	if (Om <= -2.7e-132)
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	elseif (Om <= 1.5e-100)
		tmp = sqrt(((2.0 * (U * ((l * n) * ((l * -2.0) + ((U_42_ * (l * n)) / Om))))) / Om));
	else
		tmp = sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -2.7e-132], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(l * N[(n * N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.5e-100], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(l * n), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(U$42$ * N[(l * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(N[(N[(l * n), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -2.6999999999999999e-132

   1. Initial program 53.2%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 52.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 59.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 59.4%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(\ell \cdot n\right) \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\ell \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \left(\frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\ell \cdot U\right), Om\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 62.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, U\right), Om\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 62.2%

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

   if -2.6999999999999999e-132 < Om < 1.5e-100

   1. Initial program 43.3%

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

   3. Simplified 60.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 65.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot 2\right) \cdot U\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot U\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot 2\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot \left(U \cdot 2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot n\right) \cdot \left(U \cdot 2\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right), \left(n \cdot \left(U \cdot 2\right)\right)\right)\right) \]

   8. Applied egg-rr 60.0%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       4. *-lowering-*.f64 60.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

   11. Simplified 60.0%

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

   12. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\left(\ell \cdot -2\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right)\right), Om\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right)\right), Om\right)\right) \]

       13. *-lowering-*.f64 67.6%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right)\right), Om\right)\right) \]

   14. Simplified 67.6%

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

   if 1.5e-100 < Om

   1. Initial program 47.8%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 47.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 47.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 51.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.7 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.5 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot -2 + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot n\right) \cdot \left(U \cdot 2\right)\\ \mathbf{if}\;Om \leq -2.9 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\frac{-2 \cdot \left(U \cdot \left(\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* t n) (* U 2.0))))
   (if (<= Om -2.9e-134)
     (sqrt (+ t_1 (* -4.0 (* l (* n (/ (* U l) Om))))))
     (if (<= Om 1.65e-103)
       (sqrt (/ (* -2.0 (* U (* (- 2.0 (/ (* U* n) Om)) (* n (* l l))))) Om))
       (sqrt (+ t_1 (* -4.0 (/ (* (* l n) (* U l)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -2.9e-134) {
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 1.65e-103) {
		tmp = sqrt(((-2.0 * (U * ((2.0 - ((U_42_ * n) / Om)) * (n * (l * l))))) / Om));
	} else {
		tmp = sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * n) * (u * 2.0d0)
    if (om <= (-2.9d-134)) then
        tmp = sqrt((t_1 + ((-4.0d0) * (l * (n * ((u * l) / om))))))
    else if (om <= 1.65d-103) then
        tmp = sqrt((((-2.0d0) * (u * ((2.0d0 - ((u_42 * n) / om)) * (n * (l * l))))) / om))
    else
        tmp = sqrt((t_1 + ((-4.0d0) * (((l * n) * (u * l)) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -2.9e-134) {
		tmp = Math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 1.65e-103) {
		tmp = Math.sqrt(((-2.0 * (U * ((2.0 - ((U_42_ * n) / Om)) * (n * (l * l))))) / Om));
	} else {
		tmp = Math.sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (t * n) * (U * 2.0)
	tmp = 0
	if Om <= -2.9e-134:
		tmp = math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))))
	elif Om <= 1.65e-103:
		tmp = math.sqrt(((-2.0 * (U * ((2.0 - ((U_42_ * n) / Om)) * (n * (l * l))))) / Om))
	else:
		tmp = math.sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(t * n) * Float64(U * 2.0))
	tmp = 0.0
	if (Om <= -2.9e-134)
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(l * Float64(n * Float64(Float64(U * l) / Om))))));
	elseif (Om <= 1.65e-103)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(U * Float64(Float64(2.0 - Float64(Float64(U_42_ * n) / Om)) * Float64(n * Float64(l * l))))) / Om));
	else
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(Float64(Float64(l * n) * Float64(U * l)) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (t * n) * (U * 2.0);
	tmp = 0.0;
	if (Om <= -2.9e-134)
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	elseif (Om <= 1.65e-103)
		tmp = sqrt(((-2.0 * (U * ((2.0 - ((U_42_ * n) / Om)) * (n * (l * l))))) / Om));
	else
		tmp = sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -2.9e-134], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(l * N[(n * N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.65e-103], N[Sqrt[N[(N[(-2.0 * N[(U * N[(N[(2.0 - N[(N[(U$42$ * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(N[(N[(l * n), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -2.89999999999999993e-134

   1. Initial program 53.2%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 52.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 59.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 59.4%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(\ell \cdot n\right) \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\ell \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \left(\frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\ell \cdot U\right), Om\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 62.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, U\right), Om\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 62.2%

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

   if -2.89999999999999993e-134 < Om < 1.64999999999999995e-103

   1. Initial program 41.6%

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

   3. Simplified 58.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 64.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot 2\right) \cdot U\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot U\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot 2\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot \left(U \cdot 2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot n\right) \cdot \left(U \cdot 2\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right), \left(n \cdot \left(U \cdot 2\right)\right)\right)\right) \]

   8. Applied egg-rr 58.8%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

       4. *-lowering-*.f64 58.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right) \]

   11. Simplified 58.8%

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

   12. Taylor expanded in l around -inf

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   14. Simplified 60.7%

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

   if 1.64999999999999995e-103 < Om

   1. Initial program 49.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 46.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 50.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 50.4%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.9 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\frac{-2 \cdot \left(U \cdot \left(\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot n\right) \cdot \left(U \cdot 2\right)\\ \mathbf{if}\;Om \leq -1.12 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.9 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* t n) (* U 2.0))))
   (if (<= Om -1.12e-172)
     (sqrt (+ t_1 (* -4.0 (* l (* n (/ (* U l) Om))))))
     (if (<= Om 3.9e-103)
       (sqrt (* n (/ (* 2.0 (* U (* U* (* n (* l l))))) (* Om Om))))
       (sqrt (+ t_1 (* -4.0 (/ (* (* l n) (* U l)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -1.12e-172) {
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 3.9e-103) {
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	} else {
		tmp = sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * n) * (u * 2.0d0)
    if (om <= (-1.12d-172)) then
        tmp = sqrt((t_1 + ((-4.0d0) * (l * (n * ((u * l) / om))))))
    else if (om <= 3.9d-103) then
        tmp = sqrt((n * ((2.0d0 * (u * (u_42 * (n * (l * l))))) / (om * om))))
    else
        tmp = sqrt((t_1 + ((-4.0d0) * (((l * n) * (u * l)) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -1.12e-172) {
		tmp = Math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 3.9e-103) {
		tmp = Math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	} else {
		tmp = Math.sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (t * n) * (U * 2.0)
	tmp = 0
	if Om <= -1.12e-172:
		tmp = math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))))
	elif Om <= 3.9e-103:
		tmp = math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))))
	else:
		tmp = math.sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(t * n) * Float64(U * 2.0))
	tmp = 0.0
	if (Om <= -1.12e-172)
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(l * Float64(n * Float64(Float64(U * l) / Om))))));
	elseif (Om <= 3.9e-103)
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l * l))))) / Float64(Om * Om))));
	else
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(Float64(Float64(l * n) * Float64(U * l)) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (t * n) * (U * 2.0);
	tmp = 0.0;
	if (Om <= -1.12e-172)
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	elseif (Om <= 3.9e-103)
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	else
		tmp = sqrt((t_1 + (-4.0 * (((l * n) * (U * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.12e-172], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(l * N[(n * N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 3.9e-103], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(N[(N[(l * n), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.11999999999999996e-172

   1. Initial program 52.7%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 58.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 58.5%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(\ell \cdot n\right) \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\ell \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \left(\frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\ell \cdot U\right), Om\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, U\right), Om\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 61.0%

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

   if -1.11999999999999996e-172 < Om < 3.9000000000000002e-103

   1. Initial program 42.2%

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

   3. Simplified 59.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 65.5%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}, n\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)}{{Om}^{2}}\right), n\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left(Om \cdot Om\right)\right), n\right)\right) \]

      10. *-lowering-*.f64 55.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), n\right)\right) \]

   9. Simplified 55.8%

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

   if 3.9000000000000002e-103 < Om

   1. Initial program 48.4%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 46.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 46.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 50.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 50.9%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.12 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.9 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \frac{\left(\ell \cdot n\right) \cdot \left(U \cdot \ell\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot n\right) \cdot \left(U \cdot 2\right)\\ \mathbf{if}\;Om \leq -1.65 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* t n) (* U 2.0))))
   (if (<= Om -1.65e-165)
     (sqrt (+ t_1 (* -4.0 (* l (* n (/ (* U l) Om))))))
     (if (<= Om 2.8e-102)
       (sqrt (* n (/ (* 2.0 (* U (* U* (* n (* l l))))) (* Om Om))))
       (sqrt (+ t_1 (* -4.0 (* (* l (* l n)) (/ U Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -1.65e-165) {
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 2.8e-102) {
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	} else {
		tmp = sqrt((t_1 + (-4.0 * ((l * (l * n)) * (U / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * n) * (u * 2.0d0)
    if (om <= (-1.65d-165)) then
        tmp = sqrt((t_1 + ((-4.0d0) * (l * (n * ((u * l) / om))))))
    else if (om <= 2.8d-102) then
        tmp = sqrt((n * ((2.0d0 * (u * (u_42 * (n * (l * l))))) / (om * om))))
    else
        tmp = sqrt((t_1 + ((-4.0d0) * ((l * (l * n)) * (u / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t * n) * (U * 2.0);
	double tmp;
	if (Om <= -1.65e-165) {
		tmp = Math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	} else if (Om <= 2.8e-102) {
		tmp = Math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	} else {
		tmp = Math.sqrt((t_1 + (-4.0 * ((l * (l * n)) * (U / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (t * n) * (U * 2.0)
	tmp = 0
	if Om <= -1.65e-165:
		tmp = math.sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))))
	elif Om <= 2.8e-102:
		tmp = math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))))
	else:
		tmp = math.sqrt((t_1 + (-4.0 * ((l * (l * n)) * (U / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(t * n) * Float64(U * 2.0))
	tmp = 0.0
	if (Om <= -1.65e-165)
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(l * Float64(n * Float64(Float64(U * l) / Om))))));
	elseif (Om <= 2.8e-102)
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l * l))))) / Float64(Om * Om))));
	else
		tmp = sqrt(Float64(t_1 + Float64(-4.0 * Float64(Float64(l * Float64(l * n)) * Float64(U / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (t * n) * (U * 2.0);
	tmp = 0.0;
	if (Om <= -1.65e-165)
		tmp = sqrt((t_1 + (-4.0 * (l * (n * ((U * l) / Om))))));
	elseif (Om <= 2.8e-102)
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	else
		tmp = sqrt((t_1 + (-4.0 * ((l * (l * n)) * (U / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.65e-165], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(l * N[(n * N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.8e-102], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 + N[(-4.0 * N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.6499999999999999e-165

   1. Initial program 52.7%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 58.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 58.5%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(\ell \cdot n\right) \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\ell \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \left(\frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\ell \cdot U\right), Om\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, U\right), Om\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 61.0%

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

   if -1.6499999999999999e-165 < Om < 2.80000000000000013e-102

   1. Initial program 42.2%

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

   3. Simplified 59.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 65.5%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}, n\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)}{{Om}^{2}}\right), n\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left(Om \cdot Om\right)\right), n\right)\right) \]

      10. *-lowering-*.f64 55.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), n\right)\right) \]

   9. Simplified 55.8%

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

   if 2.80000000000000013e-102 < Om

   1. Initial program 48.4%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 46.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 46.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(n \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{U}{Om}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 50.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(U, Om\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 50.5%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.65 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -1.25 \cdot 10^{-167}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 2.65 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt (+ (* (* t n) (* U 2.0)) (* -4.0 (* l (* n (/ (* U l) Om))))))))
   (if (<= Om -1.25e-167)
     t_1
     (if (<= Om 2.65e-102)
       (sqrt (* n (/ (* 2.0 (* U (* U* (* n (* l l))))) (* Om Om))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((t * n) * (U * 2.0)) + (-4.0 * (l * (n * ((U * l) / Om))))));
	double tmp;
	if (Om <= -1.25e-167) {
		tmp = t_1;
	} else if (Om <= 2.65e-102) {
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((t * n) * (u * 2.0d0)) + ((-4.0d0) * (l * (n * ((u * l) / om))))))
    if (om <= (-1.25d-167)) then
        tmp = t_1
    else if (om <= 2.65d-102) then
        tmp = sqrt((n * ((2.0d0 * (u * (u_42 * (n * (l * l))))) / (om * om))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * (l * (n * ((U * l) / Om))))));
	double tmp;
	if (Om <= -1.25e-167) {
		tmp = t_1;
	} else if (Om <= 2.65e-102) {
		tmp = Math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((t * n) * (U * 2.0)) + (-4.0 * (l * (n * ((U * l) / Om))))))
	tmp = 0
	if Om <= -1.25e-167:
		tmp = t_1
	elif Om <= 2.65e-102:
		tmp = math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(t * n) * Float64(U * 2.0)) + Float64(-4.0 * Float64(l * Float64(n * Float64(Float64(U * l) / Om))))))
	tmp = 0.0
	if (Om <= -1.25e-167)
		tmp = t_1;
	elseif (Om <= 2.65e-102)
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l * l))))) / Float64(Om * Om))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((t * n) * (U * 2.0)) + (-4.0 * (l * (n * ((U * l) / Om))))));
	tmp = 0.0;
	if (Om <= -1.25e-167)
		tmp = t_1;
	elseif (Om <= 2.65e-102)
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(l * N[(n * N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -1.25e-167], t$95$1, If[LessEqual[Om, 2.65e-102], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.25000000000000005e-167 or 2.6500000000000001e-102 < Om

   1. Initial program 50.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 49.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 49.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 55.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 55.2%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\left(\ell \cdot n\right) \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\ell \cdot \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \left(n \cdot \frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \left(\frac{\ell \cdot U}{Om}\right)\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\ell \cdot U\right), Om\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 56.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, U\right), Om\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 56.2%

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

   if -1.25000000000000005e-167 < Om < 2.6500000000000001e-102

   1. Initial program 42.2%

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

   3. Simplified 59.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 65.5%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}, n\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)}{{Om}^{2}}\right), n\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left(Om \cdot Om\right)\right), n\right)\right) \]

      10. *-lowering-*.f64 55.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), n\right)\right) \]

   9. Simplified 55.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.25 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.65 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right) + -4 \cdot \left(\ell \cdot \left(n \cdot \frac{U \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(t \cdot n\right)\\ \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(t\_1 \cdot t\_1\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* t n))))
   (if (<= l 5.6e+104)
     (sqrt (* (* t n) (* U 2.0)))
     (if (<= l 8.5e+200)
       (sqrt (/ (* -4.0 (* n (* U (* l l)))) Om))
       (pow (* 4.0 (* t_1 t_1)) 0.25)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (t * n);
	double tmp;
	if (l <= 5.6e+104) {
		tmp = sqrt(((t * n) * (U * 2.0)));
	} else if (l <= 8.5e+200) {
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	} else {
		tmp = pow((4.0 * (t_1 * t_1)), 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (t * n)
    if (l <= 5.6d+104) then
        tmp = sqrt(((t * n) * (u * 2.0d0)))
    else if (l <= 8.5d+200) then
        tmp = sqrt((((-4.0d0) * (n * (u * (l * l)))) / om))
    else
        tmp = (4.0d0 * (t_1 * t_1)) ** 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (t * n);
	double tmp;
	if (l <= 5.6e+104) {
		tmp = Math.sqrt(((t * n) * (U * 2.0)));
	} else if (l <= 8.5e+200) {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	} else {
		tmp = Math.pow((4.0 * (t_1 * t_1)), 0.25);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U * (t * n)
	tmp = 0
	if l <= 5.6e+104:
		tmp = math.sqrt(((t * n) * (U * 2.0)))
	elif l <= 8.5e+200:
		tmp = math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om))
	else:
		tmp = math.pow((4.0 * (t_1 * t_1)), 0.25)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(t * n))
	tmp = 0.0
	if (l <= 5.6e+104)
		tmp = sqrt(Float64(Float64(t * n) * Float64(U * 2.0)));
	elseif (l <= 8.5e+200)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l * l)))) / Om));
	else
		tmp = Float64(4.0 * Float64(t_1 * t_1)) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (t * n);
	tmp = 0.0;
	if (l <= 5.6e+104)
		tmp = sqrt(((t * n) * (U * 2.0)));
	elseif (l <= 8.5e+200)
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	else
		tmp = (4.0 * (t_1 * t_1)) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(t * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5.6e+104], N[Sqrt[N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.5e+200], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Power[N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.6e104

   1. Initial program 53.3%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 42.8%

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

   if 5.6e104 < l < 8.5e200

   1. Initial program 36.7%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 46.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 58.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 58.6%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\left(U \cdot {\ell}^{2}\right) \cdot n\right)\right), Om\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(U \cdot {\ell}^{2}\right), n\right)\right), Om\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right), n\right)\right), Om\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right), n\right)\right), Om\right)\right) \]

       8. *-lowering-*.f64 31.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), n\right)\right), Om\right)\right) \]

   12. Simplified 31.9%

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

   if 8.5e200 < l

   1. Initial program 7.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 13.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(n \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(U \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot U\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot t\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 13.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right)\right) \]

   9. Applied egg-rr 13.2%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

       8. metadata-eval N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)} \]

       9. metadata-eval N/A

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

       10. pow-prod-up N/A

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

       11. pow-prod-down N/A

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

       12. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 19.3%

       \[\leadsto \color{blue}{{\left(4 \cdot \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)}^{0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.4e+161)
   (sqrt (* (* U 2.0) (* n (- t (* 2.0 (/ (* l l) Om))))))
   (sqrt (* n (/ (* 2.0 (* U (* U* (* n (* l l))))) (* Om Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.4e+161) {
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.4d+161) then
        tmp = sqrt(((u * 2.0d0) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * (u_42 * (n * (l * l))))) / (om * om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.4e+161) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.4e+161:
		tmp = math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.4e+161)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l * l))))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.4e+161)
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l * l))))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.4e+161], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.40000000000000004e161

   1. Initial program 52.0%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

   if 6.40000000000000004e161 < l

   1. Initial program 18.5%

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

   3. Simplified 47.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 46.6%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}, n\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)}{{Om}^{2}}\right), n\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left({Om}^{2}\right)\right), n\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \left(Om \cdot Om\right)\right), n\right)\right) \]

      10. *-lowering-*.f64 26.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), n\right)\right) \]

   9. Simplified 26.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e+161)
   (sqrt (* (* U 2.0) (* n (- t (* 2.0 (/ (* l l) Om))))))
   (sqrt (* (* U (* 2.0 n)) (/ (* U* (* n (* l l))) (* Om Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e+161) {
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt(((U * (2.0 * n)) * ((U_42_ * (n * (l * l))) / (Om * Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2d+161) then
        tmp = sqrt(((u * 2.0d0) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt(((u * (2.0d0 * n)) * ((u_42 * (n * (l * l))) / (om * om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e+161) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt(((U * (2.0 * n)) * ((U_42_ * (n * (l * l))) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2e+161:
		tmp = math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt(((U * (2.0 * n)) * ((U_42_ * (n * (l * l))) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2e+161)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(U_42_ * Float64(n * Float64(l * l))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2e+161)
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt(((U * (2.0 * n)) * ((U_42_ * (n * (l * l))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2e+161], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.0000000000000001e161

   1. Initial program 52.0%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

   if 2.0000000000000001e161 < l

   1. Initial program 18.5%

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

   3. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(n \cdot {\ell}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right) \]

      8. *-lowering-*.f64 26.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right) \]

   5. Simplified 26.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.2e+161)
   (sqrt (* (* U 2.0) (* n (- t (* 2.0 (/ (* l l) Om))))))
   (sqrt (* 2.0 (* U (/ (* (* U* (* l l)) (* n n)) (* Om Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.2e+161) {
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((2.0 * (U * (((U_42_ * (l * l)) * (n * n)) / (Om * Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.2d+161) then
        tmp = sqrt(((u * 2.0d0) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((2.0d0 * (u * (((u_42 * (l * l)) * (n * n)) / (om * om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.2e+161) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (((U_42_ * (l * l)) * (n * n)) / (Om * Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.2e+161:
		tmp = math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((2.0 * (U * (((U_42_ * (l * l)) * (n * n)) / (Om * Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.2e+161)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(Float64(U_42_ * Float64(l * l)) * Float64(n * n)) / Float64(Om * Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.2e+161)
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((2.0 * (U * (((U_42_ * (l * l)) * (n * n)) / (Om * Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.2e+161], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2e161

   1. Initial program 52.0%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

   if 2.2e161 < l

   1. Initial program 18.5%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{{Om}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\left(\left(U* \cdot {\ell}^{2}\right) \cdot {n}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U* \cdot {\ell}^{2}\right), \left({n}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2}\right)\right), \left({n}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \ell\right)\right), \left({n}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({n}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(n \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, n\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 25.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right) \]

   7. Simplified 25.4%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \left(2 \cdot t\right)\right)\\ \mathbf{if}\;\ell \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_1 \cdot t\_1\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U (* 2.0 t)))))
   (if (<= l 5e+200)
     (sqrt (* (* U 2.0) (* n (- t (* 2.0 (/ (* l l) Om))))))
     (pow (* t_1 t_1) 0.25))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * (2.0 * t));
	double tmp;
	if (l <= 5e+200) {
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = pow((t_1 * t_1), 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (u * (2.0d0 * t))
    if (l <= 5d+200) then
        tmp = sqrt(((u * 2.0d0) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = (t_1 * t_1) ** 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * (2.0 * t));
	double tmp;
	if (l <= 5e+200) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.pow((t_1 * t_1), 0.25);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U * (2.0 * t))
	tmp = 0
	if l <= 5e+200:
		tmp = math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.pow((t_1 * t_1), 0.25)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * Float64(2.0 * t)))
	tmp = 0.0
	if (l <= 5e+200)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = Float64(t_1 * t_1) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U * (2.0 * t));
	tmp = 0.0;
	if (l <= 5e+200)
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = (t_1 * t_1) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5e+200], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000019e200

   1. Initial program 51.9%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

   if 5.00000000000000019e200 < l

   1. Initial program 7.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 13.1%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right) \cdot \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(n \cdot U\right)\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(U \cdot n\right)\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot U\right), n\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot t\right), U\right), n\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \left(\left(2 \cdot t\right) \cdot \left(n \cdot U\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \left(\left(2 \cdot t\right) \cdot \left(U \cdot n\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot U\right), n\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot t\right), U\right), n\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

      18. metadata-eval 19.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right)\right), \frac{1}{4}\right) \]

   9. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{{\left(\left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right) \cdot \left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right)\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot \left(U \cdot \left(2 \cdot t\right)\right)\right) \cdot \left(n \cdot \left(U \cdot \left(2 \cdot t\right)\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(t \cdot n\right)\\ \mathbf{if}\;\ell \leq 8.8 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(t\_1 \cdot t\_1\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* t n))))
   (if (<= l 8.8e+200)
     (sqrt (* (* U 2.0) (* n (- t (* 2.0 (/ (* l l) Om))))))
     (pow (* 4.0 (* t_1 t_1)) 0.25))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (t * n);
	double tmp;
	if (l <= 8.8e+200) {
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = pow((4.0 * (t_1 * t_1)), 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (t * n)
    if (l <= 8.8d+200) then
        tmp = sqrt(((u * 2.0d0) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = (4.0d0 * (t_1 * t_1)) ** 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (t * n);
	double tmp;
	if (l <= 8.8e+200) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.pow((4.0 * (t_1 * t_1)), 0.25);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U * (t * n)
	tmp = 0
	if l <= 8.8e+200:
		tmp = math.sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.pow((4.0 * (t_1 * t_1)), 0.25)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(t * n))
	tmp = 0.0
	if (l <= 8.8e+200)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = Float64(4.0 * Float64(t_1 * t_1)) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (t * n);
	tmp = 0.0;
	if (l <= 8.8e+200)
		tmp = sqrt(((U * 2.0) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = (4.0 * (t_1 * t_1)) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(t * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 8.8e+200], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 8.8e200

   1. Initial program 51.9%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 52.4%

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

   if 8.8e200 < l

   1. Initial program 7.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 13.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(n \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(U \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot U\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot t\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 13.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right)\right) \]

   9. Applied egg-rr 13.2%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

       8. metadata-eval N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)} \]

       9. metadata-eval N/A

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

       10. pow-prod-up N/A

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

       11. pow-prod-down N/A

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

       12. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 19.3%

       \[\leadsto \color{blue}{{\left(4 \cdot \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.4e+104)
   (sqrt (* (* t n) (* U 2.0)))
   (sqrt (/ (* -4.0 (* n (* U (* l l)))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.4e+104) {
		tmp = sqrt(((t * n) * (U * 2.0)));
	} else {
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 5.4d+104) then
        tmp = sqrt(((t * n) * (u * 2.0d0)))
    else
        tmp = sqrt((((-4.0d0) * (n * (u * (l * l)))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.4e+104) {
		tmp = Math.sqrt(((t * n) * (U * 2.0)));
	} else {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.4e+104:
		tmp = math.sqrt(((t * n) * (U * 2.0)))
	else:
		tmp = math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.4e+104)
		tmp = sqrt(Float64(Float64(t * n) * Float64(U * 2.0)));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l * l)))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.4e+104)
		tmp = sqrt(((t * n) * (U * 2.0)));
	else
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.4e+104], N[Sqrt[N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.39999999999999969e104

   1. Initial program 53.3%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 42.8%

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

   if 5.39999999999999969e104 < l

   1. Initial program 23.0%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 28.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 28.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right), Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U\right), Om\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot U\right)\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 48.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, U\right)\right), Om\right)\right)\right)\right) \]

   9. Applied egg-rr 48.1%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\left(U \cdot {\ell}^{2}\right) \cdot n\right)\right), Om\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(U \cdot {\ell}^{2}\right), n\right)\right), Om\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right), n\right)\right), Om\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right), n\right)\right), Om\right)\right) \]

       8. *-lowering-*.f64 21.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), n\right)\right), Om\right)\right) \]

   12. Simplified 21.0%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.6e+104)
   (sqrt (* (* t n) (* U 2.0)))
   (sqrt (/ (* -4.0 (* U (* n (* l l)))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.6e+104) {
		tmp = sqrt(((t * n) * (U * 2.0)));
	} else {
		tmp = sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 5.6d+104) then
        tmp = sqrt(((t * n) * (u * 2.0d0)))
    else
        tmp = sqrt((((-4.0d0) * (u * (n * (l * l)))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.6e+104) {
		tmp = Math.sqrt(((t * n) * (U * 2.0)));
	} else {
		tmp = Math.sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.6e+104:
		tmp = math.sqrt(((t * n) * (U * 2.0)))
	else:
		tmp = math.sqrt(((-4.0 * (U * (n * (l * l)))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.6e+104)
		tmp = sqrt(Float64(Float64(t * n) * Float64(U * 2.0)));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(U * Float64(n * Float64(l * l)))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.6e+104)
		tmp = sqrt(((t * n) * (U * 2.0)));
	else
		tmp = sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.6e+104], N[Sqrt[N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.6e104

   1. Initial program 53.3%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 42.8%

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

   if 5.6e104 < l

   1. Initial program 23.0%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right)\right) \]

      13. *-lowering-*.f64 28.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 28.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 23.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right) \]

   10. Simplified 23.4%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -2.25 \cdot 10^{-297}:\\ \;\;\;\;{\left(n \cdot \left(U \cdot \left(2 \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -2.25e-297)
   (pow (* n (* U (* 2.0 t))) 0.5)
   (sqrt (* (* t n) (* U 2.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.25e-297) {
		tmp = pow((n * (U * (2.0 * t))), 0.5);
	} else {
		tmp = sqrt(((t * n) * (U * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= (-2.25d-297)) then
        tmp = (n * (u * (2.0d0 * t))) ** 0.5d0
    else
        tmp = sqrt(((t * n) * (u * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.25e-297) {
		tmp = Math.pow((n * (U * (2.0 * t))), 0.5);
	} else {
		tmp = Math.sqrt(((t * n) * (U * 2.0)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -2.25e-297:
		tmp = math.pow((n * (U * (2.0 * t))), 0.5)
	else:
		tmp = math.sqrt(((t * n) * (U * 2.0)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -2.25e-297)
		tmp = Float64(n * Float64(U * Float64(2.0 * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(t * n) * Float64(U * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -2.25e-297)
		tmp = (n * (U * (2.0 * t))) ^ 0.5;
	else
		tmp = sqrt(((t * n) * (U * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -2.25e-297], N[Power[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.24999999999999988e-297

   1. Initial program 53.4%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 40.3%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(n \cdot U\right)\right), \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot t\right) \cdot \left(U \cdot n\right)\right), \frac{1}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot t\right) \cdot U\right), n\right), \frac{1}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot t\right), U\right), n\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot t\right) \cdot U\right) \cdot n\right)}^{0.5}} \]

   if -2.24999999999999988e-297 < U*

   1. Initial program 44.4%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 32.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 32.9%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.25 \cdot 10^{-297}:\\ \;\;\;\;{\left(n \cdot \left(U \cdot \left(2 \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 3.5e-216)
   (sqrt (* (* t n) (* U 2.0)))
   (sqrt (* t (* U (* 2.0 n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 3.5e-216) {
		tmp = sqrt(((t * n) * (U * 2.0)));
	} else {
		tmp = sqrt((t * (U * (2.0 * n))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 3.5d-216) then
        tmp = sqrt(((t * n) * (u * 2.0d0)))
    else
        tmp = sqrt((t * (u * (2.0d0 * n))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 3.5e-216) {
		tmp = Math.sqrt(((t * n) * (U * 2.0)));
	} else {
		tmp = Math.sqrt((t * (U * (2.0 * n))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 3.5e-216:
		tmp = math.sqrt(((t * n) * (U * 2.0)))
	else:
		tmp = math.sqrt((t * (U * (2.0 * n))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 3.5e-216)
		tmp = sqrt(Float64(Float64(t * n) * Float64(U * 2.0)));
	else
		tmp = sqrt(Float64(t * Float64(U * Float64(2.0 * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= 3.5e-216)
		tmp = sqrt(((t * n) * (U * 2.0)));
	else
		tmp = sqrt((t * (U * (2.0 * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 3.5e-216], N[Sqrt[N[(N[(t * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 3.49999999999999982e-216

   1. Initial program 49.6%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 41.4%

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

   if 3.49999999999999982e-216 < Om

   1. Initial program 47.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 38.2%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 35.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Simplified 59.0%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 38.4%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

7. Simplified 38.4%

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

8. Final simplification 38.4%

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

Alternative 25: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Simplified 59.0%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 35.9%

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

8. Final simplification 35.9%

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

Reproduce

herbie shell --seed 2024138 
(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*))))))