Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 65.1%

Time: 34.8s

Alternatives: 21

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, t_1, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot {\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(U - U*\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 1e-156)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (fma (- U U*) t_1 (* (* l l) (/ 2.0 Om)))))))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* 2.0 (* n U))
         (pow (cbrt (- t (fma 2.0 (* l (/ l Om)) (* t_1 (- U U*))))) 3.0)))
       (sqrt
        (*
         (* l (* (* n l) (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om))))
         (* U -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 1e-156) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma((U - U_42_), t_1, ((l * l) * (2.0 / Om))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * pow(cbrt((t - fma(2.0, (l * (l / Om)), (t_1 * (U - U_42_))))), 3.0)));
	} else {
		tmp = sqrt(((l * ((n * l) * fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)))) * (U * -2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 1e-156)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(Float64(U - U_42_), t_1, Float64(Float64(l * l) * Float64(2.0 / Om)))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * (cbrt(Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(U - U_42_))))) ^ 3.0)));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)))) * Float64(U * -2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e-156], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(U - U$42$), $MachinePrecision] * t$95$1 + N[(N[(l * l), $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000004e-156

   1. Initial program 18.9%

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

   3. Simplified 18.9%

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

      1. pow1/2 18.9%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5}} \]

      2. fma-udef 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)}^{0.5} \]

      3. associate-*l/ 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5} \]

      4. associate-*r* 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)\right)}^{0.5} \]

      5. *-commutative 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)\right)}^{0.5} \]

      6. associate--l- 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{0.5} \]

      7. associate-*r* 18.9%

         \[\leadsto {\left(\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}^{0.5} \]

      8. associate-*l* 35.1%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}}^{0.5} \]

   5. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. *-commutative 55.1%

         \[\leadsto \sqrt{\color{blue}{n \cdot 2}} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5} \]

      2. unpow1/2 55.1%

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

      3. fma-udef 55.1%

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

      4. associate-*r/ 55.1%

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

      5. unpow2 55.1%

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

      6. associate-*r/ 55.1%

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

      7. unpow2 55.1%

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

      8. +-commutative 55.1%

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

      9. unpow2 55.1%

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

      10. associate-*r/ 55.1%

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

   7. Simplified 55.1%

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

   if 1.00000000000000004e-156 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

      1. metadata-eval 70.2%

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

      2. cancel-sign-sub-inv 70.2%

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

      3. associate-*r* 70.2%

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

      4. add-cube-cbrt 69.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 69.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 73.6%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. metadata-eval 0.0%

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

      2. cancel-sign-sub-inv 0.0%

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

      3. associate-*r* 0.0%

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

      4. add-cube-cbrt 0.0%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 0.0%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 7.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 37.2%

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

      1. associate-*r* 37.2%

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

      2. associate-*r* 37.3%

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

      3. unpow2 37.3%

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

      4. *-commutative 37.3%

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

      5. unpow2 37.3%

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

      6. times-frac 48.4%

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

      7. associate-*r/ 48.4%

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

      8. metadata-eval 48.4%

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

      9. *-commutative 48.4%

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

   8. Simplified 48.4%

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

   9. Taylor expanded in l around 0 37.2%

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

       1. associate-*r* 37.3%

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

       2. unpow2 37.3%

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

       3. associate-*r* 46.0%

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

       4. associate-*r/ 46.0%

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

       5. metadata-eval 46.0%

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

       6. unpow2 46.0%

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

       7. times-frac 60.6%

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

       8. associate-*l* 62.3%

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

       9. metadata-eval 62.3%

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

       10. associate-*r/ 62.3%

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

       11. +-commutative 62.3%

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

       12. fma-def 62.3%

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

       13. associate-*r/ 62.3%

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

       14. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot {\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \]

Alternative 2: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left|\left(t - \mathsf{fma}\left(U - U*, t_1, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 0.0)
     (sqrt (* (* 2.0 U) (* n (fma l (* (/ l Om) -2.0) t))))
     (if (<= t_2 5e+141)
       (sqrt
        (fabs
         (* (- t (fma (- U U*) t_1 (* (* l l) (/ 2.0 Om)))) (* 2.0 (* n U)))))
       (sqrt
        (*
         (* l (* (* n l) (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om))))
         (* U -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((2.0 * U) * (n * fma(l, ((l / Om) * -2.0), t))));
	} else if (t_2 <= 5e+141) {
		tmp = sqrt(fabs(((t - fma((U - U_42_), t_1, ((l * l) * (2.0 / Om)))) * (2.0 * (n * U)))));
	} else {
		tmp = sqrt(((l * ((n * l) * fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)))) * (U * -2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * fma(l, Float64(Float64(l / Om) * -2.0), t))));
	elseif (t_2 <= 5e+141)
		tmp = sqrt(abs(Float64(Float64(t - fma(Float64(U - U_42_), t_1, Float64(Float64(l * l) * Float64(2.0 / Om)))) * Float64(2.0 * Float64(n * U)))));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)))) * Float64(U * -2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+141], N[Sqrt[N[Abs[N[(N[(t - N[(N[(U - U$42$), $MachinePrecision] * t$95$1 + N[(N[(l * l), $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in n around 0 33.1%

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

      1. associate-*r* 33.1%

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

      2. cancel-sign-sub-inv 33.1%

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

      3. metadata-eval 33.1%

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

      4. +-commutative 33.1%

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

      5. unpow2 33.1%

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

      6. associate-*r/ 33.1%

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

      7. *-commutative 33.1%

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

      8. associate-*l* 33.1%

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

      9. fma-def 33.1%

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

   6. Simplified 33.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000025e141

   1. Initial program 95.8%

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

   3. Simplified 88.7%

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

      1. associate-*r* 88.7%

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

      2. fma-udef 88.7%

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

      3. associate-*l/ 88.7%

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

      4. associate-*r* 95.8%

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

      5. *-commutative 95.8%

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

      6. associate--l- 95.8%

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

      7. add-sqr-sqrt 95.8%

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

   5. Applied egg-rr 56.8%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{2}\right)}^{0.5}}} \]
   6. Step-by-step derivation

      1. unpow1/2 56.8%

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

      2. unpow2 56.8%

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

      3. rem-sqrt-square 95.9%

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

      4. fma-udef 95.9%

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

      5. associate-*r/ 95.8%

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

      6. unpow2 95.8%

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

      7. associate-*r/ 95.8%

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

      8. unpow2 95.8%

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

      9. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   if 5.00000000000000025e141 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Simplified 22.0%

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

      1. metadata-eval 22.0%

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

      2. cancel-sign-sub-inv 22.0%

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

      3. associate-*r* 21.3%

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

      4. add-cube-cbrt 21.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 21.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 29.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 34.6%

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

      1. associate-*r* 34.6%

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

      2. associate-*r* 34.7%

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

      3. unpow2 34.7%

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

      4. *-commutative 34.7%

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

      5. unpow2 34.7%

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

      6. times-frac 39.6%

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

      7. associate-*r/ 39.6%

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

      8. metadata-eval 39.6%

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

      9. *-commutative 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in l around 0 34.6%

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

       1. associate-*r* 34.7%

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

       2. unpow2 34.7%

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

       3. associate-*r* 40.2%

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

       4. associate-*r/ 40.2%

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

       5. metadata-eval 40.2%

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

       6. unpow2 40.2%

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

       7. times-frac 46.5%

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

       8. associate-*l* 50.3%

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

       9. metadata-eval 50.3%

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

       10. associate-*r/ 50.3%

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

       11. +-commutative 50.3%

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

       12. fma-def 50.3%

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

       13. associate-*r/ 50.3%

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

       14. metadata-eval 50.3%

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

   11. Simplified 50.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left|\left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \]

Alternative 3: 63.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := t - \mathsf{fma}\left(U - U*, t_1, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t_2}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left|t_2 \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2 (- t (fma (- U U*) t_1 (* (* l l) (/ 2.0 Om)))))
        (t_3
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_3 1e-156)
     (* (sqrt (* 2.0 n)) (sqrt (* U t_2)))
     (if (<= t_3 5e+141)
       (sqrt (fabs (* t_2 (* 2.0 (* n U)))))
       (sqrt
        (*
         (* l (* (* n l) (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om))))
         (* U -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = t - fma((U - U_42_), t_1, ((l * l) * (2.0 / Om)));
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 1e-156) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t_2));
	} else if (t_3 <= 5e+141) {
		tmp = sqrt(fabs((t_2 * (2.0 * (n * U)))));
	} else {
		tmp = sqrt(((l * ((n * l) * fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)))) * (U * -2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = Float64(t - fma(Float64(U - U_42_), t_1, Float64(Float64(l * l) * Float64(2.0 / Om))))
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 1e-156)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t_2)));
	elseif (t_3 <= 5e+141)
		tmp = sqrt(abs(Float64(t_2 * Float64(2.0 * Float64(n * U)))));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)))) * Float64(U * -2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(N[(U - U$42$), $MachinePrecision] * t$95$1 + N[(N[(l * l), $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-156], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+141], N[Sqrt[N[Abs[N[(t$95$2 * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000004e-156

   1. Initial program 18.9%

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

   3. Simplified 18.9%

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

      1. pow1/2 18.9%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5}} \]

      2. fma-udef 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)}^{0.5} \]

      3. associate-*l/ 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5} \]

      4. associate-*r* 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\right)\right)\right)}^{0.5} \]

      5. *-commutative 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)\right)}^{0.5} \]

      6. associate--l- 18.9%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{0.5} \]

      7. associate-*r* 18.9%

         \[\leadsto {\left(\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}^{0.5} \]

      8. associate-*l* 35.1%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}}^{0.5} \]

   5. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. *-commutative 55.1%

         \[\leadsto \sqrt{\color{blue}{n \cdot 2}} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5} \]

      2. unpow1/2 55.1%

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

      3. fma-udef 55.1%

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

      4. associate-*r/ 55.1%

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

      5. unpow2 55.1%

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

      6. associate-*r/ 55.1%

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

      7. unpow2 55.1%

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

      8. +-commutative 55.1%

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

      9. unpow2 55.1%

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

      10. associate-*r/ 55.1%

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

   7. Simplified 55.1%

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

   if 1.00000000000000004e-156 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000025e141

   1. Initial program 96.5%

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

   3. Simplified 89.2%

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

      1. associate-*r* 89.2%

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

      2. fma-udef 89.2%

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

      3. associate-*l/ 89.2%

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

      4. associate-*r* 96.5%

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

      5. *-commutative 96.5%

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

      6. associate--l- 96.5%

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

      7. add-sqr-sqrt 96.5%

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

   5. Applied egg-rr 57.9%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{2}\right)}^{0.5}}} \]
   6. Step-by-step derivation

      1. unpow1/2 57.9%

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

      2. unpow2 57.9%

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

      3. rem-sqrt-square 96.5%

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

      4. fma-udef 96.5%

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

      5. associate-*r/ 96.5%

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

      6. unpow2 96.5%

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

      7. associate-*r/ 96.5%

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

      8. unpow2 96.5%

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

      9. +-commutative 96.5%

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

   7. Simplified 96.5%

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

   if 5.00000000000000025e141 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Simplified 22.0%

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

      1. metadata-eval 22.0%

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

      2. cancel-sign-sub-inv 22.0%

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

      3. associate-*r* 21.3%

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

      4. add-cube-cbrt 21.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 21.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 29.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 34.6%

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

      1. associate-*r* 34.6%

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

      2. associate-*r* 34.7%

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

      3. unpow2 34.7%

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

      4. *-commutative 34.7%

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

      5. unpow2 34.7%

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

      6. times-frac 39.6%

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

      7. associate-*r/ 39.6%

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

      8. metadata-eval 39.6%

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

      9. *-commutative 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in l around 0 34.6%

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

       1. associate-*r* 34.7%

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

       2. unpow2 34.7%

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

       3. associate-*r* 40.2%

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

       4. associate-*r/ 40.2%

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

       5. metadata-eval 40.2%

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

       6. unpow2 40.2%

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

       7. times-frac 46.5%

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

       8. associate-*l* 50.3%

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

       9. metadata-eval 50.3%

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

       10. associate-*r/ 50.3%

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

       11. +-commutative 50.3%

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

       12. fma-def 50.3%

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

       13. associate-*r/ 50.3%

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

       14. metadata-eval 50.3%

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

   11. Simplified 50.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left|\left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \]

Alternative 4: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (sqrt (* (* 2.0 U) (* n (fma l (* (/ l Om) -2.0) t))))
     (if (<= t_1 5e+141)
       t_1
       (sqrt
        (*
         (* l (* (* n l) (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om))))
         (* U -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((2.0 * U) * (n * fma(l, ((l / Om) * -2.0), t))));
	} else if (t_1 <= 5e+141) {
		tmp = t_1;
	} else {
		tmp = sqrt(((l * ((n * l) * fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)))) * (U * -2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * fma(l, Float64(Float64(l / Om) * -2.0), t))));
	elseif (t_1 <= 5e+141)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)))) * Float64(U * -2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+141], t$95$1, N[Sqrt[N[(N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in n around 0 33.1%

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

      1. associate-*r* 33.1%

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

      2. cancel-sign-sub-inv 33.1%

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

      3. metadata-eval 33.1%

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

      4. +-commutative 33.1%

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

      5. unpow2 33.1%

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

      6. associate-*r/ 33.1%

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

      7. *-commutative 33.1%

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

      8. associate-*l* 33.1%

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

      9. fma-def 33.1%

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

   6. Simplified 33.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000025e141

   1. Initial program 95.8%

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

   if 5.00000000000000025e141 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Simplified 22.0%

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

      1. metadata-eval 22.0%

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

      2. cancel-sign-sub-inv 22.0%

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

      3. associate-*r* 21.3%

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

      4. add-cube-cbrt 21.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 21.3%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 29.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 34.6%

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

      1. associate-*r* 34.6%

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

      2. associate-*r* 34.7%

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

      3. unpow2 34.7%

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

      4. *-commutative 34.7%

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

      5. unpow2 34.7%

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

      6. times-frac 39.6%

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

      7. associate-*r/ 39.6%

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

      8. metadata-eval 39.6%

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

      9. *-commutative 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in l around 0 34.6%

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

       1. associate-*r* 34.7%

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

       2. unpow2 34.7%

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

       3. associate-*r* 40.2%

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

       4. associate-*r/ 40.2%

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

       5. metadata-eval 40.2%

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

       6. unpow2 40.2%

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

       7. times-frac 46.5%

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

       8. associate-*l* 50.3%

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

       9. metadata-eval 50.3%

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

       10. associate-*r/ 50.3%

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

       11. +-commutative 50.3%

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

       12. fma-def 50.3%

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

       13. associate-*r/ 50.3%

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

       14. metadata-eval 50.3%

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

   11. Simplified 50.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \]

Alternative 5: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 6e-68)
     (sqrt
      (*
       t_1
       (-
        (+ t (* (/ (* l l) Om) -2.0))
        (* n (* (- U U*) (pow (/ l Om) 2.0))))))
     (if (<= l 7.5e+79)
       (sqrt
        (* t_1 (+ t (* (* l l) (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om))))))
       (sqrt
        (*
         (* l (* (* n l) (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om))))
         (* U -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 6e-68) {
		tmp = sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) - (n * ((U - U_42_) * pow((l / Om), 2.0))))));
	} else if (l <= 7.5e+79) {
		tmp = sqrt((t_1 * (t + ((l * l) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))))));
	} else {
		tmp = sqrt(((l * ((n * l) * fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)))) * (U * -2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 6e-68)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) - Float64(n * Float64(Float64(U - U_42_) * (Float64(l / Om) ^ 2.0))))));
	elseif (l <= 7.5e+79)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))))));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)))) * Float64(U * -2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 6e-68], N[Sqrt[N[(t$95$1 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - N[(n * N[(N[(U - U$42$), $MachinePrecision] * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.5e+79], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6e-68

   1. Initial program 55.5%

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

   3. Simplified 55.6%

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

   if 6e-68 < l < 7.49999999999999967e79

   1. Initial program 42.7%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in l around 0 51.2%

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

      1. unpow2 51.2%

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

      2. associate-*r/ 51.2%

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

      3. metadata-eval 51.2%

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

      4. *-commutative 51.2%

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

      5. unpow2 51.2%

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

      6. times-frac 62.3%

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

   6. Simplified 62.3%

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

   if 7.49999999999999967e79 < l

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

      1. metadata-eval 24.2%

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

      2. cancel-sign-sub-inv 24.2%

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

      3. associate-*r* 24.2%

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

      4. add-cube-cbrt 24.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 24.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 40.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 44.0%

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

      1. associate-*r* 44.0%

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

      2. associate-*r* 44.0%

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

      3. unpow2 44.0%

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

      4. *-commutative 44.0%

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

      5. unpow2 44.0%

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

      6. times-frac 49.0%

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

      7. associate-*r/ 49.0%

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

      8. metadata-eval 49.0%

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

      9. *-commutative 49.0%

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

   8. Simplified 49.0%

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

   9. Taylor expanded in l around 0 44.0%

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

       1. associate-*r* 44.0%

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

       2. unpow2 44.0%

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

       3. associate-*r* 50.7%

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

       4. associate-*r/ 50.7%

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

       5. metadata-eval 50.7%

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

       6. unpow2 50.7%

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

       7. times-frac 57.9%

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

       8. associate-*l* 62.4%

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

       9. metadata-eval 62.4%

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

       10. associate-*r/ 62.4%

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

       11. +-commutative 62.4%

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

       12. fma-def 62.4%

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

       13. associate-*r/ 62.4%

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

       14. metadata-eval 62.4%

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

   11. Simplified 62.4%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) - n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \]

Alternative 6: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 5.4e-68)
     (sqrt (* t_1 (+ t (* (/ n Om) (/ l (/ Om (* l U*)))))))
     (if (<= l 5.6e+79)
       (sqrt
        (* t_1 (+ t (* (* l l) (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om))))))
       (sqrt
        (*
         (* l (* (* n l) (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om))))
         (* U -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 5.4e-68) {
		tmp = sqrt((t_1 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else if (l <= 5.6e+79) {
		tmp = sqrt((t_1 * (t + ((l * l) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))))));
	} else {
		tmp = sqrt(((l * ((n * l) * fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)))) * (U * -2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 5.4e-68)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(n / Om) * Float64(l / Float64(Om / Float64(l * U_42_)))))));
	elseif (l <= 5.6e+79)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))))));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)))) * Float64(U * -2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5.4e-68], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(l / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.6e+79], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.4000000000000003e-68

   1. Initial program 55.5%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in l around 0 46.5%

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

      1. unpow2 46.5%

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

      2. associate-*r/ 46.5%

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

      3. metadata-eval 46.5%

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

      4. *-commutative 46.5%

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

      5. unpow2 46.5%

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

      6. times-frac 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in U* around inf 47.1%

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

      1. mul-1-neg 47.1%

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

      2. associate-*r* 46.6%

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

      3. *-commutative 46.6%

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

      4. unpow2 46.6%

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

      5. associate-*r* 48.3%

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

      6. unpow2 48.3%

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

      7. times-frac 56.2%

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

      8. *-commutative 56.2%

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

      9. associate-/l* 58.2%

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

   9. Simplified 58.2%

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

   if 5.4000000000000003e-68 < l < 5.6000000000000002e79

   1. Initial program 42.7%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in l around 0 51.2%

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

      1. unpow2 51.2%

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

      2. associate-*r/ 51.2%

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

      3. metadata-eval 51.2%

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

      4. *-commutative 51.2%

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

      5. unpow2 51.2%

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

      6. times-frac 62.3%

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

   6. Simplified 62.3%

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

   if 5.6000000000000002e79 < l

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

      1. metadata-eval 24.2%

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

      2. cancel-sign-sub-inv 24.2%

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

      3. associate-*r* 24.2%

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

      4. add-cube-cbrt 24.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 24.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 40.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 44.0%

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

      1. associate-*r* 44.0%

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

      2. associate-*r* 44.0%

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

      3. unpow2 44.0%

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

      4. *-commutative 44.0%

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

      5. unpow2 44.0%

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

      6. times-frac 49.0%

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

      7. associate-*r/ 49.0%

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

      8. metadata-eval 49.0%

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

      9. *-commutative 49.0%

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

   8. Simplified 49.0%

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

   9. Taylor expanded in l around 0 44.0%

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

       1. associate-*r* 44.0%

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

       2. unpow2 44.0%

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

       3. associate-*r* 50.7%

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

       4. associate-*r/ 50.7%

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

       5. metadata-eval 50.7%

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

       6. unpow2 50.7%

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

       7. times-frac 57.9%

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

       8. associate-*l* 62.4%

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

       9. metadata-eval 62.4%

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

       10. associate-*r/ 62.4%

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

       11. +-commutative 62.4%

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

       12. fma-def 62.4%

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

       13. associate-*r/ 62.4%

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

       14. metadata-eval 62.4%

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

   11. Simplified 62.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)\right) \cdot \left(U \cdot -2\right)}\\ \end{array} \]

Alternative 7: 54.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n}{Om} \cdot \frac{U* - U}{Om}\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 6.9 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(t_1 - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\frac{2}{Om} - t_1\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (/ n Om) (/ (- U* U) Om))) (t_2 (* 2.0 (* n U))))
   (if (<= l 6.9e-68)
     (sqrt (* t_2 (+ t (* (/ n Om) (/ l (/ Om (* l U*)))))))
     (if (<= l 2.05e+137)
       (sqrt (* t_2 (+ t (* (* l l) (- t_1 (/ 2.0 Om))))))
       (sqrt (* (* U -2.0) (* (- (/ 2.0 Om) t_1) (* l (* n l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n / Om) * ((U_42_ - U) / Om);
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (l <= 6.9e-68) {
		tmp = sqrt((t_2 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else if (l <= 2.05e+137) {
		tmp = sqrt((t_2 * (t + ((l * l) * (t_1 - (2.0 / Om))))));
	} else {
		tmp = sqrt(((U * -2.0) * (((2.0 / Om) - t_1) * (l * (n * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (n / om) * ((u_42 - u) / om)
    t_2 = 2.0d0 * (n * u)
    if (l <= 6.9d-68) then
        tmp = sqrt((t_2 * (t + ((n / om) * (l / (om / (l * u_42)))))))
    else if (l <= 2.05d+137) then
        tmp = sqrt((t_2 * (t + ((l * l) * (t_1 - (2.0d0 / om))))))
    else
        tmp = sqrt(((u * (-2.0d0)) * (((2.0d0 / om) - t_1) * (l * (n * 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 t_1 = (n / Om) * ((U_42_ - U) / Om);
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (l <= 6.9e-68) {
		tmp = Math.sqrt((t_2 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else if (l <= 2.05e+137) {
		tmp = Math.sqrt((t_2 * (t + ((l * l) * (t_1 - (2.0 / Om))))));
	} else {
		tmp = Math.sqrt(((U * -2.0) * (((2.0 / Om) - t_1) * (l * (n * l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n / Om) * ((U_42_ - U) / Om)
	t_2 = 2.0 * (n * U)
	tmp = 0
	if l <= 6.9e-68:
		tmp = math.sqrt((t_2 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))))
	elif l <= 2.05e+137:
		tmp = math.sqrt((t_2 * (t + ((l * l) * (t_1 - (2.0 / Om))))))
	else:
		tmp = math.sqrt(((U * -2.0) * (((2.0 / Om) - t_1) * (l * (n * l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))
	t_2 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 6.9e-68)
		tmp = sqrt(Float64(t_2 * Float64(t + Float64(Float64(n / Om) * Float64(l / Float64(Om / Float64(l * U_42_)))))));
	elseif (l <= 2.05e+137)
		tmp = sqrt(Float64(t_2 * Float64(t + Float64(Float64(l * l) * Float64(t_1 - Float64(2.0 / Om))))));
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(Float64(2.0 / Om) - t_1) * Float64(l * Float64(n * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n / Om) * ((U_42_ - U) / Om);
	t_2 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 6.9e-68)
		tmp = sqrt((t_2 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	elseif (l <= 2.05e+137)
		tmp = sqrt((t_2 * (t + ((l * l) * (t_1 - (2.0 / Om))))));
	else
		tmp = sqrt(((U * -2.0) * (((2.0 / Om) - t_1) * (l * (n * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 6.9e-68], N[Sqrt[N[(t$95$2 * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(l / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.05e+137], N[Sqrt[N[(t$95$2 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(t$95$1 - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(N[(2.0 / Om), $MachinePrecision] - t$95$1), $MachinePrecision] * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.90000000000000031e-68

   1. Initial program 55.5%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in l around 0 46.5%

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

      1. unpow2 46.5%

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

      2. associate-*r/ 46.5%

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

      3. metadata-eval 46.5%

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

      4. *-commutative 46.5%

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

      5. unpow2 46.5%

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

      6. times-frac 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in U* around inf 47.1%

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

      1. mul-1-neg 47.1%

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

      2. associate-*r* 46.6%

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

      3. *-commutative 46.6%

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

      4. unpow2 46.6%

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

      5. associate-*r* 48.3%

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

      6. unpow2 48.3%

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

      7. times-frac 56.2%

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

      8. *-commutative 56.2%

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

      9. associate-/l* 58.2%

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

   9. Simplified 58.2%

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

   if 6.90000000000000031e-68 < l < 2.04999999999999998e137

   1. Initial program 43.8%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in l around 0 50.0%

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

      1. unpow2 50.0%

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

      2. associate-*r/ 50.0%

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

      3. metadata-eval 50.0%

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

      4. *-commutative 50.0%

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

      5. unpow2 50.0%

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

      6. times-frac 58.0%

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

   6. Simplified 58.0%

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

   if 2.04999999999999998e137 < l

   1. Initial program 14.6%

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

   3. Simplified 14.6%

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

      1. metadata-eval 14.6%

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

      2. cancel-sign-sub-inv 14.6%

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

      3. associate-*r* 14.6%

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

      4. add-cube-cbrt 14.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 14.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 37.3%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 39.2%

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

      1. associate-*r* 39.2%

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

      2. associate-*r* 39.2%

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

      3. unpow2 39.2%

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

      4. *-commutative 39.2%

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

      5. unpow2 39.2%

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

      6. times-frac 46.2%

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

      7. associate-*r/ 46.2%

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

      8. metadata-eval 46.2%

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

      9. *-commutative 46.2%

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

   8. Simplified 46.2%

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

      1. *-un-lft-identity 46.2%

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

      2. *-commutative 46.2%

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

      3. associate-*l* 59.0%

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

   10. Applied egg-rr 59.0%

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

       1. *-lft-identity 59.0%

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

   12. Simplified 59.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.9 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 8: 52.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.5e+75)
   (sqrt (* (* 2.0 (* n U)) (+ t (* (/ n Om) (/ l (/ Om (* l U*)))))))
   (sqrt
    (*
     (* U -2.0)
     (* (- (/ 2.0 Om) (* (/ n Om) (/ (- U* U) Om))) (* l (* n l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.5e+75) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else {
		tmp = sqrt(((U * -2.0) * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (l * (n * 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 (l <= 7.5d+75) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n / om) * (l / (om / (l * u_42)))))))
    else
        tmp = sqrt(((u * (-2.0d0)) * (((2.0d0 / om) - ((n / om) * ((u_42 - u) / om))) * (l * (n * 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 (l <= 7.5e+75) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else {
		tmp = Math.sqrt(((U * -2.0) * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (l * (n * l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7.5e+75:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))))
	else:
		tmp = math.sqrt(((U * -2.0) * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (l * (n * l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.5e+75)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n / Om) * Float64(l / Float64(Om / Float64(l * U_42_)))))));
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))) * Float64(l * Float64(n * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 7.5e+75)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	else
		tmp = sqrt(((U * -2.0) * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (l * (n * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.5e+75], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(l / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.4999999999999995e75

   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 51.6%

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

   4. Taylor expanded in l around 0 46.9%

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

      1. unpow2 46.9%

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

      2. associate-*r/ 46.9%

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

      3. metadata-eval 46.9%

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

      4. *-commutative 46.9%

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

      5. unpow2 46.9%

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

      6. times-frac 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in U* around inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. associate-*r* 45.6%

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

      3. *-commutative 45.6%

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

      4. unpow2 45.6%

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

      5. associate-*r* 47.1%

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

      6. unpow2 47.1%

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

      7. times-frac 54.5%

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

      8. *-commutative 54.5%

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

      9. associate-/l* 56.7%

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

   9. Simplified 56.7%

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

   if 7.4999999999999995e75 < l

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

      1. metadata-eval 26.0%

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

      2. cancel-sign-sub-inv 26.0%

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

      3. associate-*r* 26.0%

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

      4. add-cube-cbrt 25.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}} \]

      5. pow3 25.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)}^{3}}} \]

   5. Applied egg-rr 41.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Taylor expanded in l around inf 44.4%

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

      1. associate-*r* 44.4%

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

      2. associate-*r* 44.4%

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

      3. unpow2 44.4%

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

      4. *-commutative 44.4%

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

      5. unpow2 44.4%

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

      6. times-frac 49.3%

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

      7. associate-*r/ 49.3%

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

      8. metadata-eval 49.3%

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

      9. *-commutative 49.3%

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

   8. Simplified 49.3%

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

      1. *-un-lft-identity 49.3%

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

      2. *-commutative 49.3%

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

      3. associate-*l* 58.0%

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

   10. Applied egg-rr 58.0%

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

       1. *-lft-identity 58.0%

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

   12. Simplified 58.0%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 9: 52.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 5e+76)
     (sqrt (* t_1 (+ t (* (/ n Om) (/ l (/ Om (* l U*)))))))
     (sqrt (* t_1 (* (* l (/ l Om)) (+ -2.0 (* n (/ (- U* U) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 5e+76) {
		tmp = sqrt((t_1 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else {
		tmp = sqrt((t_1 * ((l * (l / Om)) * (-2.0 + (n * ((U_42_ - 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 = 2.0d0 * (n * u)
    if (l <= 5d+76) then
        tmp = sqrt((t_1 * (t + ((n / om) * (l / (om / (l * u_42)))))))
    else
        tmp = sqrt((t_1 * ((l * (l / om)) * ((-2.0d0) + (n * ((u_42 - 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 = 2.0 * (n * U);
	double tmp;
	if (l <= 5e+76) {
		tmp = Math.sqrt((t_1 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else {
		tmp = Math.sqrt((t_1 * ((l * (l / Om)) * (-2.0 + (n * ((U_42_ - U) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if l <= 5e+76:
		tmp = math.sqrt((t_1 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))))
	else:
		tmp = math.sqrt((t_1 * ((l * (l / Om)) * (-2.0 + (n * ((U_42_ - U) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 5e+76)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(n / Om) * Float64(l / Float64(Om / Float64(l * U_42_)))))));
	else
		tmp = sqrt(Float64(t_1 * Float64(Float64(l * Float64(l / Om)) * Float64(-2.0 + Float64(n * Float64(Float64(U_42_ - U) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 5e+76)
		tmp = sqrt((t_1 * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	else
		tmp = sqrt((t_1 * ((l * (l / Om)) * (-2.0 + (n * ((U_42_ - U) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5e+76], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(l / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(-2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.99999999999999991e76

   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 51.6%

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

   4. Taylor expanded in l around 0 46.9%

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

      1. unpow2 46.9%

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

      2. associate-*r/ 46.9%

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

      3. metadata-eval 46.9%

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

      4. *-commutative 46.9%

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

      5. unpow2 46.9%

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

      6. times-frac 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in U* around inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. associate-*r* 45.6%

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

      3. *-commutative 45.6%

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

      4. unpow2 45.6%

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

      5. associate-*r* 47.1%

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

      6. unpow2 47.1%

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

      7. times-frac 54.5%

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

      8. *-commutative 54.5%

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

      9. associate-/l* 56.7%

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

   9. Simplified 56.7%

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

   if 4.99999999999999991e76 < l

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in t around 0 23.3%

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

      1. associate-*r* 23.3%

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

      2. unpow2 23.3%

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

      3. associate-*r/ 23.3%

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

      4. *-commutative 23.3%

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

      5. associate-*l* 23.3%

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

      6. unpow2 23.3%

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

      7. unpow2 23.3%

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

   6. Simplified 23.3%

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

      1. *-un-lft-identity 23.3%

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

      2. times-frac 27.7%

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

      3. associate-*r/ 32.5%

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

      4. *-commutative 32.5%

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

   8. Applied egg-rr 32.5%

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

      1. *-lft-identity 32.5%

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

      2. associate-*r* 34.0%

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

      3. associate-*r* 34.0%

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

      4. *-commutative 34.0%

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

      5. associate-*r* 34.0%

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

      6. distribute-lft-out-- 48.1%

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

      7. associate-*l/ 48.1%

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

      8. *-commutative 48.1%

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

   10. Simplified 48.1%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)}\\ \end{array} \]

Alternative 10: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{n}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.5e+59)
   (sqrt (* 2.0 (* (* n U) (+ t (* (/ n Om) (* (/ l Om) (* l U*)))))))
   (pow (* -4.0 (/ U (/ Om (* n (* l l))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.5e+59) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((n / Om) * ((l / Om) * (l * U_42_)))))));
	} else {
		tmp = pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	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 <= 1.5d+59) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((n / om) * ((l / om) * (l * u_42)))))))
    else
        tmp = ((-4.0d0) * (u / (om / (n * (l * l))))) ** 0.5d0
    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 <= 1.5e+59) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((n / Om) * ((l / Om) * (l * U_42_)))))));
	} else {
		tmp = Math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.5e+59:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((n / Om) * ((l / Om) * (l * U_42_)))))))
	else:
		tmp = math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.5e+59)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(n / Om) * Float64(Float64(l / Om) * Float64(l * U_42_)))))));
	else
		tmp = Float64(-4.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.5e+59)
		tmp = sqrt((2.0 * ((n * U) * (t + ((n / Om) * ((l / Om) * (l * U_42_)))))));
	else
		tmp = (-4.0 * (U / (Om / (n * (l * l))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.5e+59], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5e59

   1. Initial program 53.6%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in l around 0 47.1%

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

      1. unpow2 47.1%

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

      2. associate-*r/ 47.1%

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

      3. metadata-eval 47.1%

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

      4. *-commutative 47.1%

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

      5. unpow2 47.1%

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

      6. times-frac 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in U* around inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. associate-*r* 45.8%

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

      3. *-commutative 45.8%

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

      4. unpow2 45.8%

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

      5. associate-*r* 47.3%

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

      6. unpow2 47.3%

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

      7. times-frac 54.8%

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

      8. *-commutative 54.8%

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

      9. associate-/l* 57.0%

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

   9. Simplified 57.0%

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

       1. *-un-lft-identity 57.0%

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

       2. distribute-lft-neg-in 57.0%

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

       3. associate-/r/ 57.0%

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

   11. Applied egg-rr 57.0%

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

       1. *-lft-identity 57.0%

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

       2. associate-*l* 57.0%

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

       3. cancel-sign-sub 57.0%

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

   13. Simplified 57.0%

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

   if 1.5e59 < l

   1. Initial program 26.6%

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

   3. Simplified 26.6%

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

   4. Taylor expanded in t around 0 21.9%

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

      1. associate-*r* 21.9%

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

      2. unpow2 21.9%

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

      3. associate-*r/ 21.9%

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

      4. *-commutative 21.9%

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

      5. associate-*l* 21.9%

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

      6. unpow2 21.9%

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

      7. unpow2 21.9%

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

   6. Simplified 21.9%

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

   7. Taylor expanded in n around 0 29.1%

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

      1. pow1/2 50.7%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 50.7%

         \[\leadsto {\left(-4 \cdot \color{blue}{\frac{U}{\frac{Om}{{\ell}^{2} \cdot n}}}\right)}^{0.5} \]

      3. pow2 50.7%

         \[\leadsto {\left(-4 \cdot \frac{U}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}}\right)}^{0.5} \]

   9. Applied egg-rr 50.7%

      \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{n}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 11: 50.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.5e+56)
   (sqrt (* (* 2.0 (* n U)) (+ t (* (/ n Om) (/ l (/ Om (* l U*)))))))
   (pow (* -4.0 (/ U (/ Om (* n (* l l))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.5e+56) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else {
		tmp = pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	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 <= 6.5d+56) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n / om) * (l / (om / (l * u_42)))))))
    else
        tmp = ((-4.0d0) * (u / (om / (n * (l * l))))) ** 0.5d0
    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 <= 6.5e+56) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	} else {
		tmp = Math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.5e+56:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))))
	else:
		tmp = math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.5e+56)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n / Om) * Float64(l / Float64(Om / Float64(l * U_42_)))))));
	else
		tmp = Float64(-4.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.5e+56)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / Om) * (l / (Om / (l * U_42_)))))));
	else
		tmp = (-4.0 * (U / (Om / (n * (l * l))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.5e+56], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(l / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.5000000000000001e56

   1. Initial program 53.6%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in l around 0 47.1%

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

      1. unpow2 47.1%

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

      2. associate-*r/ 47.1%

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

      3. metadata-eval 47.1%

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

      4. *-commutative 47.1%

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

      5. unpow2 47.1%

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

      6. times-frac 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in U* around inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. associate-*r* 45.8%

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

      3. *-commutative 45.8%

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

      4. unpow2 45.8%

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

      5. associate-*r* 47.3%

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

      6. unpow2 47.3%

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

      7. times-frac 54.8%

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

      8. *-commutative 54.8%

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

      9. associate-/l* 57.0%

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

   9. Simplified 57.0%

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

   if 6.5000000000000001e56 < l

   1. Initial program 26.6%

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

   3. Simplified 26.6%

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

   4. Taylor expanded in t around 0 21.9%

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

      1. associate-*r* 21.9%

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

      2. unpow2 21.9%

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

      3. associate-*r/ 21.9%

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

      4. *-commutative 21.9%

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

      5. associate-*l* 21.9%

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

      6. unpow2 21.9%

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

      7. unpow2 21.9%

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

   6. Simplified 21.9%

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

   7. Taylor expanded in n around 0 29.1%

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

      1. pow1/2 50.7%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 50.7%

         \[\leadsto {\left(-4 \cdot \color{blue}{\frac{U}{\frac{Om}{{\ell}^{2} \cdot n}}}\right)}^{0.5} \]

      3. pow2 50.7%

         \[\leadsto {\left(-4 \cdot \frac{U}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}}\right)}^{0.5} \]

   9. Applied egg-rr 50.7%

      \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{\ell}{\frac{Om}{\ell \cdot U*}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 12: 48.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+105}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -2.15e+105)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (pow (* (* 2.0 (* n U)) (- t (/ 2.0 (/ Om (* l l))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2.15e+105) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = pow(((2.0 * (n * U)) * (t - (2.0 / (Om / (l * l))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-2.15d+105)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = ((2.0d0 * (n * u)) * (t - (2.0d0 / (om / (l * l))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2.15e+105) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * (n * U)) * (t - (2.0 / (Om / (l * l))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -2.15e+105:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.pow(((2.0 * (n * U)) * (t - (2.0 / (Om / (l * l))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -2.15e+105)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(2.0 / Float64(Om / Float64(l * l))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.15e+105)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = ((2.0 * (n * U)) * (t - (2.0 / (Om / (l * l))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -2.15e+105], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1500000000000001e105

   1. Initial program 33.9%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in t around inf 54.9%

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

      1. pow1/2 58.0%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. *-commutative 58.0%

         \[\leadsto {\left(2 \cdot \left(U \cdot \color{blue}{\left(t \cdot n\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)}^{0.5}} \]

   if -2.1500000000000001e105 < t

   1. Initial program 50.9%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in Om around inf 42.9%

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

      1. associate-*r/ 42.9%

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

      2. unpow2 42.9%

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

   6. Simplified 42.9%

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

      1. pow1/2 51.1%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}^{0.5}} \]

      2. associate-/l* 51.1%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\frac{2}{\frac{Om}{\ell \cdot \ell}}}\right)\right)}^{0.5} \]

   8. Applied egg-rr 51.1%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+105}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2}{\frac{Om}{\ell \cdot \ell}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 13: 46.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 9.5e+78)
   (sqrt (* (* 2.0 (* n U)) (- t (* 2.0 (/ l (/ Om l))))))
   (pow (* -4.0 (/ U (/ Om (* n (* l l))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 9.5e+78) {
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 * (l / (Om / l))))));
	} else {
		tmp = pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	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 <= 9.5d+78) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (2.0d0 * (l / (om / l))))))
    else
        tmp = ((-4.0d0) * (u / (om / (n * (l * l))))) ** 0.5d0
    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 <= 9.5e+78) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - (2.0 * (l / (Om / l))))));
	} else {
		tmp = Math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 9.5e+78:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - (2.0 * (l / (Om / l))))))
	else:
		tmp = math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 9.5e+78)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(2.0 * Float64(l / Float64(Om / l))))));
	else
		tmp = Float64(-4.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 9.5e+78)
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 * (l / (Om / l))))));
	else
		tmp = (-4.0 * (U / (Om / (n * (l * l))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 9.5e+78], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.5000000000000006e78

   1. Initial program 53.5%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in l around 0 47.2%

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

      1. unpow2 47.2%

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

      2. associate-*r/ 47.2%

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

      3. metadata-eval 47.2%

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

      4. *-commutative 47.2%

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

      5. unpow2 47.2%

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

      6. times-frac 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in Om around inf 45.2%

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

      1. unpow2 45.2%

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

      2. associate-/l* 46.5%

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

   9. Simplified 46.5%

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

   if 9.5000000000000006e78 < l

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in t around 0 22.4%

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

      1. associate-*r* 22.4%

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

      2. unpow2 22.4%

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

      3. associate-*r/ 22.4%

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

      4. *-commutative 22.4%

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

      5. associate-*l* 22.4%

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

      6. unpow2 22.4%

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

      7. unpow2 22.4%

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

   6. Simplified 22.4%

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

   7. Taylor expanded in n around 0 27.9%

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

      1. pow1/2 49.2%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 49.2%

         \[\leadsto {\left(-4 \cdot \color{blue}{\frac{U}{\frac{Om}{{\ell}^{2} \cdot n}}}\right)}^{0.5} \]

      3. pow2 49.2%

         \[\leadsto {\left(-4 \cdot \frac{U}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}}\right)}^{0.5} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 14: 44.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.2e+79)
   (sqrt (* (* 2.0 (* n U)) (- t (* (* l l) (/ 2.0 Om)))))
   (pow (* -4.0 (/ U (/ Om (* n (* l l))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.2e+79) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * (2.0 / Om)))));
	} else {
		tmp = pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	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 <= 1.2d+79) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l * l) * (2.0d0 / om)))))
    else
        tmp = ((-4.0d0) * (u / (om / (n * (l * l))))) ** 0.5d0
    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 <= 1.2e+79) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * (2.0 / Om)))));
	} else {
		tmp = Math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.2e+79:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l * l) * (2.0 / Om)))))
	else:
		tmp = math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.2e+79)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l * l) * Float64(2.0 / Om)))));
	else
		tmp = Float64(-4.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.2e+79)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l * l) * (2.0 / Om)))));
	else
		tmp = (-4.0 * (U / (Om / (n * (l * l))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.2e+79], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l * l), $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.19999999999999993e79

   1. Initial program 53.5%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in Om around inf 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-commutative 45.2%

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

      3. associate-*r/ 45.2%

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

      4. unpow2 45.2%

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

   6. Simplified 45.2%

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

   if 1.19999999999999993e79 < l

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in t around 0 22.4%

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

      1. associate-*r* 22.4%

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

      2. unpow2 22.4%

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

      3. associate-*r/ 22.4%

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

      4. *-commutative 22.4%

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

      5. associate-*l* 22.4%

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

      6. unpow2 22.4%

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

      7. unpow2 22.4%

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

   6. Simplified 22.4%

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

   7. Taylor expanded in n around 0 27.9%

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

      1. pow1/2 49.2%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 49.2%

         \[\leadsto {\left(-4 \cdot \color{blue}{\frac{U}{\frac{Om}{{\ell}^{2} \cdot n}}}\right)}^{0.5} \]

      3. pow2 49.2%

         \[\leadsto {\left(-4 \cdot \frac{U}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}}\right)}^{0.5} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \frac{2}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 15: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 960:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 960.0)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (pow (* -4.0 (/ U (/ Om (* n (* l l))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 960.0) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	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 <= 960.0d0) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = ((-4.0d0) * (u / (om / (n * (l * l))))) ** 0.5d0
    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 <= 960.0) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 960.0:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.pow((-4.0 * (U / (Om / (n * (l * l))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 960.0)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(-4.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 960.0)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = (-4.0 * (U / (Om / (n * (l * l))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 960.0], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 960

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around inf 41.3%

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

      1. pow1/2 42.8%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{0.5}} \]

      2. associate-*l* 42.8%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}}^{0.5} \]

      3. *-commutative 42.8%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)\right)}^{0.5} \]

   6. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{0.5}} \]

   if 960 < l

   1. Initial program 31.8%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in t around 0 26.7%

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

      1. associate-*r* 26.7%

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

      2. unpow2 26.7%

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

      3. associate-*r/ 26.7%

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

      4. *-commutative 26.7%

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

      5. associate-*l* 26.7%

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

      6. unpow2 26.7%

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

      7. unpow2 26.7%

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

   6. Simplified 26.7%

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

   7. Taylor expanded in n around 0 27.7%

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

      1. pow1/2 45.7%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 47.3%

         \[\leadsto {\left(-4 \cdot \color{blue}{\frac{U}{\frac{Om}{{\ell}^{2} \cdot n}}}\right)}^{0.5} \]

      3. pow2 47.3%

         \[\leadsto {\left(-4 \cdot \frac{U}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}}\right)}^{0.5} \]

   9. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot n}}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 960:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 16: 38.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+76}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.3e+76)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* -4.0 (* (* n (* l l)) (/ U Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.3e+76) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((-4.0 * ((n * (l * l)) * (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) :: tmp
    if (l <= 2.3d+76) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * ((n * (l * l)) * (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 tmp;
	if (l <= 2.3e+76) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * ((n * (l * l)) * (U / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.3e+76:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * ((n * (l * l)) * (U / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.3e+76)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(n * Float64(l * l)) * Float64(U / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.3e+76)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * ((n * (l * l)) * (U / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.3e+76], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.30000000000000001e76

   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 53.4%

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

   4. Taylor expanded in t around inf 40.3%

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

      1. pow1/2 41.8%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{0.5}} \]

      2. associate-*l* 41.8%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}}^{0.5} \]

      3. *-commutative 41.8%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)\right)}^{0.5} \]

   6. Applied egg-rr 41.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{0.5}} \]

   if 2.30000000000000001e76 < l

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in t around 0 23.3%

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

      1. associate-*r* 23.3%

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

      2. unpow2 23.3%

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

      3. associate-*r/ 23.3%

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

      4. *-commutative 23.3%

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

      5. associate-*l* 23.3%

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

      6. unpow2 23.3%

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

      7. unpow2 23.3%

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

   6. Simplified 23.3%

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

   7. Taylor expanded in n around 0 28.7%

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

      1. *-un-lft-identity 28.7%

         \[\leadsto \color{blue}{1 \cdot \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. associate-/l* 28.8%

         \[\leadsto 1 \cdot \sqrt{-4 \cdot \color{blue}{\frac{U}{\frac{Om}{{\ell}^{2} \cdot n}}}} \]

      3. pow2 28.8%

         \[\leadsto 1 \cdot \sqrt{-4 \cdot \frac{U}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}}} \]

   9. Applied egg-rr 28.8%

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

       1. *-lft-identity 28.8%

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

       2. associate-/r/ 30.8%

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

   11. Simplified 30.8%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+76}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U}{Om}\right)}\\ \end{array} \]

Alternative 17: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U}{\frac{Om}{\ell \cdot \left(n \cdot \ell\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.4e+76)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* -4.0 (/ U (/ Om (* l (* n l))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.4e+76) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((-4.0 * (U / (Om / (l * (n * 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 (l <= 2.4d+76) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * (u / (om / (l * (n * 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 (l <= 2.4e+76) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * (U / (Om / (l * (n * l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.4e+76:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * (U / (Om / (l * (n * l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.4e+76)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(U / Float64(Om / Float64(l * Float64(n * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.4e+76)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * (U / (Om / (l * (n * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.4e+76], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(U / N[(Om / N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4e76

   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 53.4%

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

   4. Taylor expanded in t around inf 40.3%

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

      1. pow1/2 41.8%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{0.5}} \]

      2. associate-*l* 41.8%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}}^{0.5} \]

      3. *-commutative 41.8%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)\right)}^{0.5} \]

   6. Applied egg-rr 41.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{0.5}} \]

   if 2.4e76 < l

   1. Initial program 26.0%

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

   3. Simplified 41.6%

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

   4. Taylor expanded in Om around inf 26.1%

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

      1. associate-*r/ 26.1%

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

      2. unpow2 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in t around 0 28.7%

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

      1. unpow2 28.7%

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

      2. associate-*r* 35.4%

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

      3. associate-/l* 35.4%

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

   9. Simplified 35.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U}{\frac{Om}{\ell \cdot \left(n \cdot \ell\right)}}}\\ \end{array} \]

Alternative 18: 36.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 1.55 \cdot 10^{-274}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 1.55e-274)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* t (* 2.0 (* n U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.55e-274) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((t * (2.0 * (n * U))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 1.55d-274) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((t * (2.0d0 * (n * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.55e-274) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 1.55e-274:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 1.55e-274)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.55e-274)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((t * (2.0 * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 1.55e-274], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.54999999999999989e-274

   1. Initial program 46.1%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in t around inf 32.0%

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

      1. pow1/2 35.1%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. *-commutative 35.1%

         \[\leadsto {\left(2 \cdot \left(U \cdot \color{blue}{\left(t \cdot n\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)}^{0.5}} \]

   if 1.54999999999999989e-274 < U

   1. Initial program 51.2%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in t around inf 42.8%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.55 \cdot 10^{-274}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 19: 37.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 3.9 \cdot 10^{-274}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 3.9e-274)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (pow (* 2.0 (* t (* n U))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 3.9e-274) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	}
	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 <= 3.9d-274) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    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 <= 3.9e-274) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 3.9e-274:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 3.9e-274)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 3.9e-274)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 3.9e-274], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 3.89999999999999985e-274

   1. Initial program 46.1%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in t around inf 32.0%

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

      1. pow1/2 35.1%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. *-commutative 35.1%

         \[\leadsto {\left(2 \cdot \left(U \cdot \color{blue}{\left(t \cdot n\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)}^{0.5}} \]

   if 3.89999999999999985e-274 < U

   1. Initial program 51.2%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in t around inf 42.8%

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

      1. pow1/2 43.6%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{0.5}} \]

      2. associate-*l* 43.6%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}}^{0.5} \]

      3. *-commutative 43.6%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot t\right)\right)}^{0.5} \]

   6. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.9 \cdot 10^{-274}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 20: 35.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -1.1e+131)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* t (* 2.0 (* n U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.1e+131) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((t * (2.0 * (n * U))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-1.1d+131)) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((t * (2.0d0 * (n * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.1e+131) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -1.1e+131:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -1.1e+131)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.1e+131)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((t * (2.0 * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -1.1e+131], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0999999999999999e131

   1. Initial program 34.8%

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

   3. Simplified 34.7%

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

   4. Taylor expanded in t around inf 57.6%

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

   if -1.0999999999999999e131 < t

   1. Initial program 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in t around inf 35.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 21: 35.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

3. Simplified 49.9%

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

4. Taylor expanded in t around inf 34.7%

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

5. Final simplification 34.7%

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

Reproduce

herbie shell --seed 2023275 
(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*))))))