Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 74.5%

Time: 25.0s

Alternatives: 18

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 74.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{{l\_m}^{2}}{Om}\\ t_2 := \sqrt{\left|n\right|}\\ t_3 := \sqrt{\left|U\right| \cdot 2}\\ \mathbf{if}\;l\_m \leq 10^{-6}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \sqrt{\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}, 2 \cdot t\_1\right)\right|}\right)\\ \mathbf{elif}\;l\_m \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \sqrt{\left|t - t\_1 \cdot \left(2 + n \cdot \frac{U - U*}{Om}\right)\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l_m 2.0) Om))
        (t_2 (sqrt (fabs n)))
        (t_3 (sqrt (* (fabs U) 2.0))))
   (if (<= l_m 1e-6)
     (*
      t_2
      (*
       t_3
       (sqrt
        (fabs (- t (fma (- U U*) (* n (pow (/ l_m Om) 2.0)) (* 2.0 t_1)))))))
     (if (<= l_m 1.2e+167)
       (*
        t_2
        (* t_3 (sqrt (fabs (- t (* t_1 (+ 2.0 (* n (/ (- U U*) Om)))))))))
       (*
        (/ (* l_m (sqrt 2.0)) Om)
        (sqrt (* U (* n (+ (* Om -2.0) (* n U*))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow(l_m, 2.0) / Om;
	double t_2 = sqrt(fabs(n));
	double t_3 = sqrt((fabs(U) * 2.0));
	double tmp;
	if (l_m <= 1e-6) {
		tmp = t_2 * (t_3 * sqrt(fabs((t - fma((U - U_42_), (n * pow((l_m / Om), 2.0)), (2.0 * t_1))))));
	} else if (l_m <= 1.2e+167) {
		tmp = t_2 * (t_3 * sqrt(fabs((t - (t_1 * (2.0 + (n * ((U - U_42_) / Om))))))));
	} else {
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64((l_m ^ 2.0) / Om)
	t_2 = sqrt(abs(n))
	t_3 = sqrt(Float64(abs(U) * 2.0))
	tmp = 0.0
	if (l_m <= 1e-6)
		tmp = Float64(t_2 * Float64(t_3 * sqrt(abs(Float64(t - fma(Float64(U - U_42_), Float64(n * (Float64(l_m / Om) ^ 2.0)), Float64(2.0 * t_1)))))));
	elseif (l_m <= 1.2e+167)
		tmp = Float64(t_2 * Float64(t_3 * sqrt(abs(Float64(t - Float64(t_1 * Float64(2.0 + Float64(n * Float64(Float64(U - U_42_) / Om)))))))));
	else
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / Om) * sqrt(Float64(U * Float64(n * Float64(Float64(Om * -2.0) + Float64(n * U_42_))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Abs[n], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Abs[U], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l$95$m, 1e-6], N[(t$95$2 * N[(t$95$3 * N[Sqrt[N[Abs[N[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.2e+167], N[(t$95$2 * N[(t$95$3 * N[Sqrt[N[Abs[N[(t - N[(t$95$1 * N[(2.0 + N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(Om * -2.0), $MachinePrecision] + N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.99999999999999955e-7

   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 56.7%

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

   6. Applied egg-rr 36.2%

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

      1. unpow1/2 36.2%

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

      2. unpow2 36.2%

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

      3. rem-sqrt-square 54.7%

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

      4. associate-*r* 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*r/ 54.8%

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

   8. Simplified 54.8%

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

      1. pow1/2 54.8%

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

      2. fabs-mul 54.8%

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

      3. unpow-prod-down 64.1%

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

      4. associate-*l* 64.1%

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

   10. Applied egg-rr 64.1%

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

       1. unpow1/2 64.1%

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

       2. unpow1/2 64.1%

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

       3. associate-*r* 64.1%

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

   12. Simplified 64.1%

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

       1. pow1/2 64.1%

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

       2. fabs-mul 64.1%

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

       3. metadata-eval 64.1%

          \[\leadsto \sqrt{\left|n\right|} \cdot {\left(\left|U \cdot 2\right| \cdot \left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. unpow-prod-down 75.2%

          \[\leadsto \sqrt{\left|n\right|} \cdot \color{blue}{\left({\left(\left|U \cdot 2\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       5. metadata-eval 75.2%

          \[\leadsto \sqrt{\left|n\right|} \cdot \left({\left(\left|U \cdot 2\right|\right)}^{\color{blue}{0.5}} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right) \]

       6. metadata-eval 75.2%

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

   14. Applied egg-rr 75.2%

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

       1. unpow1/2 75.2%

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

       2. fabs-mul 75.2%

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

       3. metadata-eval 75.2%

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

       4. unpow1/2 75.2%

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

       5. fabs-sub 75.2%

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

   16. Simplified 75.2%

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

   if 9.99999999999999955e-7 < l < 1.19999999999999999e167

   1. Initial program 39.7%

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

   3. Simplified 35.9%

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

   6. Applied egg-rr 25.8%

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

      1. unpow1/2 25.8%

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

      2. unpow2 25.8%

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

      3. rem-sqrt-square 40.0%

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

      4. associate-*r* 40.0%

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

      5. *-commutative 40.0%

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

      6. associate-*r/ 40.0%

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

   8. Simplified 40.0%

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

      1. pow1/2 40.0%

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

      2. fabs-mul 40.0%

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

      3. unpow-prod-down 53.8%

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

      4. associate-*l* 53.8%

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

   10. Applied egg-rr 53.8%

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

       1. unpow1/2 53.8%

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

       2. unpow1/2 53.8%

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

       3. associate-*r* 53.8%

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

   12. Simplified 53.8%

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

       1. pow1/2 53.8%

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

       2. fabs-mul 53.8%

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

       3. metadata-eval 53.8%

          \[\leadsto \sqrt{\left|n\right|} \cdot {\left(\left|U \cdot 2\right| \cdot \left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. unpow-prod-down 67.5%

          \[\leadsto \sqrt{\left|n\right|} \cdot \color{blue}{\left({\left(\left|U \cdot 2\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       5. metadata-eval 67.5%

          \[\leadsto \sqrt{\left|n\right|} \cdot \left({\left(\left|U \cdot 2\right|\right)}^{\color{blue}{0.5}} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right) \]

       6. metadata-eval 67.5%

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

   14. Applied egg-rr 67.5%

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

       1. unpow1/2 67.5%

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

       2. fabs-mul 67.5%

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

       3. metadata-eval 67.5%

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

       4. unpow1/2 67.5%

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

       5. fabs-sub 67.5%

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

   16. Simplified 67.5%

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

   17. Taylor expanded in U around -inf 52.7%

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

   19. Simplified 85.9%

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

   if 1.19999999999999999e167 < l

   1. Initial program 6.7%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in Om around 0 5.0%

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

   6. Taylor expanded in l around inf 41.0%

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

   7. Taylor expanded in U around 0 45.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-6}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - \frac{{\ell}^{2}}{Om} \cdot \left(2 + n \cdot \frac{U - U*}{Om}\right)\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \frac{{l\_m}^{2}}{Om}\\ t_2 := n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_3 := t\_2 \cdot \left(U* - U\right)\\ t_4 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_3\right)\\ t_5 := \sqrt{\left|n\right|}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;t\_5 \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(t - t\_1\right)\right|}\\ \mathbf{elif}\;t\_4 \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_3 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5 \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(\left(t\_1 + \left(U - U*\right) \cdot t\_2\right) - t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(-2 \cdot \left(U \cdot Om\right) + \left(U* - U\right) \cdot \left(n \cdot U\right)\right)}}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (pow l_m 2.0) Om)))
        (t_2 (* n (pow (/ l_m Om) 2.0)))
        (t_3 (* t_2 (- U* U)))
        (t_4 (* (* U (* n 2.0)) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_3)))
        (t_5 (sqrt (fabs n))))
   (if (<= t_4 0.0)
     (* t_5 (sqrt (fabs (* (* U 2.0) (- t t_1)))))
     (if (<= t_4 1e+307)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_3 (* 2.0 (* l_m (/ l_m Om)))))))
       (if (<= t_4 INFINITY)
         (* t_5 (sqrt (fabs (* (* U 2.0) (- (+ t_1 (* (- U U*) t_2)) t)))))
         (/
          (*
           (* l_m (sqrt 2.0))
           (sqrt (* n (+ (* -2.0 (* U Om)) (* (- U* U) (* n U))))))
          Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (pow(l_m, 2.0) / Om);
	double t_2 = n * pow((l_m / Om), 2.0);
	double t_3 = t_2 * (U_42_ - U);
	double t_4 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3);
	double t_5 = sqrt(fabs(n));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_5 * sqrt(fabs(((U * 2.0) * (t - t_1))));
	} else if (t_4 <= 1e+307) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * (l_m * (l_m / Om)))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5 * sqrt(fabs(((U * 2.0) * ((t_1 + ((U - U_42_) * t_2)) - t))));
	} else {
		tmp = ((l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (Math.pow(l_m, 2.0) / Om);
	double t_2 = n * Math.pow((l_m / Om), 2.0);
	double t_3 = t_2 * (U_42_ - U);
	double t_4 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3);
	double t_5 = Math.sqrt(Math.abs(n));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_5 * Math.sqrt(Math.abs(((U * 2.0) * (t - t_1))));
	} else if (t_4 <= 1e+307) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * (l_m * (l_m / Om)))))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_5 * Math.sqrt(Math.abs(((U * 2.0) * ((t_1 + ((U - U_42_) * t_2)) - t))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (math.pow(l_m, 2.0) / Om)
	t_2 = n * math.pow((l_m / Om), 2.0)
	t_3 = t_2 * (U_42_ - U)
	t_4 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3)
	t_5 = math.sqrt(math.fabs(n))
	tmp = 0
	if t_4 <= 0.0:
		tmp = t_5 * math.sqrt(math.fabs(((U * 2.0) * (t - t_1))))
	elif t_4 <= 1e+307:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * (l_m * (l_m / Om)))))))
	elif t_4 <= math.inf:
		tmp = t_5 * math.sqrt(math.fabs(((U * 2.0) * ((t_1 + ((U - U_42_) * t_2)) - t))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64((l_m ^ 2.0) / Om))
	t_2 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	t_3 = Float64(t_2 * Float64(U_42_ - U))
	t_4 = Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_3))
	t_5 = sqrt(abs(n))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(t_5 * sqrt(abs(Float64(Float64(U * 2.0) * Float64(t - t_1)))));
	elseif (t_4 <= 1e+307)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_3 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	elseif (t_4 <= Inf)
		tmp = Float64(t_5 * sqrt(abs(Float64(Float64(U * 2.0) * Float64(Float64(t_1 + Float64(Float64(U - U_42_) * t_2)) - t)))));
	else
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(n * Float64(Float64(-2.0 * Float64(U * Om)) + Float64(Float64(U_42_ - U) * Float64(n * U)))))) / Om);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * ((l_m ^ 2.0) / Om);
	t_2 = n * ((l_m / Om) ^ 2.0);
	t_3 = t_2 * (U_42_ - U);
	t_4 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3);
	t_5 = sqrt(abs(n));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = t_5 * sqrt(abs(((U * 2.0) * (t - t_1))));
	elseif (t_4 <= 1e+307)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * (l_m * (l_m / Om)))))));
	elseif (t_4 <= Inf)
		tmp = t_5 * sqrt(abs(((U * 2.0) * ((t_1 + ((U - U_42_) * t_2)) - t))));
	else
		tmp = ((l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[Abs[n], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(t$95$5 * N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(t - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+307], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$3 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$5 * N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(N[(t$95$1 + N[(N[(U - U$42$), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(N[(-2.0 * N[(U * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 43.5%

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

   6. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

      2. unpow2 26.9%

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

      3. rem-sqrt-square 46.4%

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

      4. associate-*r* 46.4%

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

      5. *-commutative 46.4%

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

      6. associate-*r/ 46.4%

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

   8. Simplified 46.4%

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

      1. pow1/2 46.4%

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

      2. fabs-mul 46.4%

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

      3. unpow-prod-down 70.2%

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

      4. associate-*l* 70.2%

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

   10. Applied egg-rr 70.2%

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

       1. unpow1/2 70.2%

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

       2. unpow1/2 70.2%

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

       3. associate-*r* 70.2%

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

   12. Simplified 70.2%

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

   13. Taylor expanded in n around 0 74.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.1%

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

   3. Simplified 98.1%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 29.0%

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

   3. Simplified 39.3%

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

   6. Applied egg-rr 29.0%

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

      1. unpow1/2 29.0%

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

      2. unpow2 29.0%

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

      3. rem-sqrt-square 31.6%

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

      4. associate-*r* 31.6%

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

      5. *-commutative 31.6%

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

      6. associate-*r/ 31.6%

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

   8. Simplified 31.6%

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

      1. pow1/2 31.6%

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

      2. fabs-mul 31.6%

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

      3. unpow-prod-down 49.6%

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

      4. associate-*l* 49.6%

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

   10. Applied egg-rr 49.6%

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

       1. unpow1/2 49.6%

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

       2. unpow1/2 49.6%

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

       3. associate-*r* 49.6%

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

   12. Simplified 49.6%

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

       1. fma-undefine 49.6%

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

   14. Applied egg-rr 49.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in Om around 0 5.4%

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

   6. Taylor expanded in l around inf 35.9%

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

      1. associate-*l/ 34.1%

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

      2. fma-define 34.1%

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

      3. mul-1-neg 34.1%

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

      4. fmm-undef 34.1%

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

      5. *-commutative 34.1%

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

      6. associate-*r* 34.1%

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

   8. Applied egg-rr 34.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) - t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(-2 \cdot \left(U \cdot Om\right) + \left(U* - U\right) \cdot \left(n \cdot U\right)\right)}}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left|n\right|}\\ t_2 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)\\ t_4 := \frac{{l\_m}^{2}}{Om}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;t\_1 \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(t - 2 \cdot t\_4\right)\right|}\\ \mathbf{elif}\;t\_3 \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(\left|U\right| \cdot 2\right) \cdot \left|t\_4 \cdot \left(2 + n \cdot \frac{U - U*}{Om}\right) - t\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (fabs n)))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3 (* (* U (* n 2.0)) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2)))
        (t_4 (/ (pow l_m 2.0) Om)))
   (if (<= t_3 0.0)
     (* t_1 (sqrt (fabs (* (* U 2.0) (- t (* 2.0 t_4))))))
     (if (<= t_3 1e+307)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        t_1
        (sqrt
         (*
          (* (fabs U) 2.0)
          (fabs (- (* t_4 (+ 2.0 (* n (/ (- U U*) Om)))) t)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(fabs(n));
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double t_4 = pow(l_m, 2.0) / Om;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = t_1 * sqrt(fabs(((U * 2.0) * (t - (2.0 * t_4)))));
	} else if (t_3 <= 1e+307) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = t_1 * sqrt(((fabs(U) * 2.0) * fabs(((t_4 * (2.0 + (n * ((U - U_42_) / Om)))) - t))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt(abs(n))
    t_2 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_3 = (u * (n * 2.0d0)) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_2)
    t_4 = (l_m ** 2.0d0) / om
    if (t_3 <= 0.0d0) then
        tmp = t_1 * sqrt(abs(((u * 2.0d0) * (t - (2.0d0 * t_4)))))
    else if (t_3 <= 1d+307) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_2 - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = t_1 * sqrt(((abs(u) * 2.0d0) * abs(((t_4 * (2.0d0 + (n * ((u - u_42) / om)))) - t))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(Math.abs(n));
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double t_4 = Math.pow(l_m, 2.0) / Om;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = t_1 * Math.sqrt(Math.abs(((U * 2.0) * (t - (2.0 * t_4)))));
	} else if (t_3 <= 1e+307) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = t_1 * Math.sqrt(((Math.abs(U) * 2.0) * Math.abs(((t_4 * (2.0 + (n * ((U - U_42_) / Om)))) - t))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(math.fabs(n))
	t_2 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)
	t_4 = math.pow(l_m, 2.0) / Om
	tmp = 0
	if t_3 <= 0.0:
		tmp = t_1 * math.sqrt(math.fabs(((U * 2.0) * (t - (2.0 * t_4)))))
	elif t_3 <= 1e+307:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = t_1 * math.sqrt(((math.fabs(U) * 2.0) * math.fabs(((t_4 * (2.0 + (n * ((U - U_42_) / Om)))) - t))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(abs(n))
	t_2 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2))
	t_4 = Float64((l_m ^ 2.0) / Om)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(t_1 * sqrt(abs(Float64(Float64(U * 2.0) * Float64(t - Float64(2.0 * t_4))))));
	elseif (t_3 <= 1e+307)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(abs(U) * 2.0) * abs(Float64(Float64(t_4 * Float64(2.0 + Float64(n * Float64(Float64(U - U_42_) / Om)))) - t)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(abs(n));
	t_2 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	t_4 = (l_m ^ 2.0) / Om;
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = t_1 * sqrt(abs(((U * 2.0) * (t - (2.0 * t_4)))));
	elseif (t_3 <= 1e+307)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = t_1 * sqrt(((abs(U) * 2.0) * abs(((t_4 * (2.0 + (n * ((U - U_42_) / Om)))) - t))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[Abs[n], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(t$95$1 * N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(t - N[(2.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+307], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] * 2.0), $MachinePrecision] * N[Abs[N[(N[(t$95$4 * N[(2.0 + N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 43.5%

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

   6. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

      2. unpow2 26.9%

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

      3. rem-sqrt-square 46.4%

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

      4. associate-*r* 46.4%

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

      5. *-commutative 46.4%

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

      6. associate-*r/ 46.4%

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

   8. Simplified 46.4%

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

      1. pow1/2 46.4%

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

      2. fabs-mul 46.4%

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

      3. unpow-prod-down 70.2%

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

      4. associate-*l* 70.2%

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

   10. Applied egg-rr 70.2%

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

       1. unpow1/2 70.2%

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

       2. unpow1/2 70.2%

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

       3. associate-*r* 70.2%

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

   12. Simplified 70.2%

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

   13. Taylor expanded in n around 0 74.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.1%

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

   3. Simplified 98.1%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.8%

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

   3. Simplified 29.7%

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

   6. Applied egg-rr 19.1%

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

      1. unpow1/2 19.1%

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

      2. unpow2 19.1%

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

      3. rem-sqrt-square 20.8%

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

      4. associate-*r* 20.8%

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

      5. *-commutative 20.8%

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

      6. associate-*r/ 20.8%

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

   8. Simplified 20.8%

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

      1. pow1/2 20.8%

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

      2. fabs-mul 20.8%

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

      3. unpow-prod-down 32.4%

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

      4. associate-*l* 32.4%

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

   10. Applied egg-rr 32.4%

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

       1. unpow1/2 32.4%

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

       2. unpow1/2 32.4%

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

       3. associate-*r* 32.4%

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

   12. Simplified 32.4%

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

       1. pow1/2 32.4%

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

       2. fabs-mul 32.4%

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

       3. metadata-eval 32.4%

          \[\leadsto \sqrt{\left|n\right|} \cdot {\left(\left|U \cdot 2\right| \cdot \left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. unpow-prod-down 39.7%

          \[\leadsto \sqrt{\left|n\right|} \cdot \color{blue}{\left({\left(\left|U \cdot 2\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       5. metadata-eval 39.7%

          \[\leadsto \sqrt{\left|n\right|} \cdot \left({\left(\left|U \cdot 2\right|\right)}^{\color{blue}{0.5}} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right) \]

       6. metadata-eval 39.7%

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

   14. Applied egg-rr 39.7%

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

       1. unpow1/2 39.7%

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

       2. fabs-mul 39.7%

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

       3. metadata-eval 39.7%

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

       4. unpow1/2 39.7%

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

       5. fabs-sub 39.7%

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

   16. Simplified 39.7%

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

       1. pow1 39.7%

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

       2. sqrt-unprod 32.4%

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

       3. fabs-sub 32.4%

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

   18. Applied egg-rr 32.4%

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

   20. Simplified 39.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \sqrt{\left(\left|U\right| \cdot 2\right) \cdot \left|\frac{{\ell}^{2}}{Om} \cdot \left(2 + n \cdot \frac{U - U*}{Om}\right) - t\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{{l\_m}^{2}}{Om}\\ t_2 := \sqrt{\left|n\right|}\\ t_3 := \sqrt{\left|U\right| \cdot 2}\\ \mathbf{if}\;l\_m \leq 1.02 \cdot 10^{-127}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \sqrt{\left|t - 2 \cdot t\_1\right|}\right)\\ \mathbf{elif}\;l\_m \leq 4.4 \cdot 10^{+166}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \sqrt{\left|t - t\_1 \cdot \left(2 + n \cdot \frac{U - U*}{Om}\right)\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l_m 2.0) Om))
        (t_2 (sqrt (fabs n)))
        (t_3 (sqrt (* (fabs U) 2.0))))
   (if (<= l_m 1.02e-127)
     (* t_2 (* t_3 (sqrt (fabs (- t (* 2.0 t_1))))))
     (if (<= l_m 4.4e+166)
       (*
        t_2
        (* t_3 (sqrt (fabs (- t (* t_1 (+ 2.0 (* n (/ (- U U*) Om)))))))))
       (*
        (/ (* l_m (sqrt 2.0)) Om)
        (sqrt (* U (* n (+ (* Om -2.0) (* n U*))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow(l_m, 2.0) / Om;
	double t_2 = sqrt(fabs(n));
	double t_3 = sqrt((fabs(U) * 2.0));
	double tmp;
	if (l_m <= 1.02e-127) {
		tmp = t_2 * (t_3 * sqrt(fabs((t - (2.0 * t_1)))));
	} else if (l_m <= 4.4e+166) {
		tmp = t_2 * (t_3 * sqrt(fabs((t - (t_1 * (2.0 + (n * ((U - U_42_) / Om))))))));
	} else {
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (l_m ** 2.0d0) / om
    t_2 = sqrt(abs(n))
    t_3 = sqrt((abs(u) * 2.0d0))
    if (l_m <= 1.02d-127) then
        tmp = t_2 * (t_3 * sqrt(abs((t - (2.0d0 * t_1)))))
    else if (l_m <= 4.4d+166) then
        tmp = t_2 * (t_3 * sqrt(abs((t - (t_1 * (2.0d0 + (n * ((u - u_42) / om))))))))
    else
        tmp = ((l_m * sqrt(2.0d0)) / om) * sqrt((u * (n * ((om * (-2.0d0)) + (n * u_42)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.pow(l_m, 2.0) / Om;
	double t_2 = Math.sqrt(Math.abs(n));
	double t_3 = Math.sqrt((Math.abs(U) * 2.0));
	double tmp;
	if (l_m <= 1.02e-127) {
		tmp = t_2 * (t_3 * Math.sqrt(Math.abs((t - (2.0 * t_1)))));
	} else if (l_m <= 4.4e+166) {
		tmp = t_2 * (t_3 * Math.sqrt(Math.abs((t - (t_1 * (2.0 + (n * ((U - U_42_) / Om))))))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) / Om) * Math.sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.pow(l_m, 2.0) / Om
	t_2 = math.sqrt(math.fabs(n))
	t_3 = math.sqrt((math.fabs(U) * 2.0))
	tmp = 0
	if l_m <= 1.02e-127:
		tmp = t_2 * (t_3 * math.sqrt(math.fabs((t - (2.0 * t_1)))))
	elif l_m <= 4.4e+166:
		tmp = t_2 * (t_3 * math.sqrt(math.fabs((t - (t_1 * (2.0 + (n * ((U - U_42_) / Om))))))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) / Om) * math.sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64((l_m ^ 2.0) / Om)
	t_2 = sqrt(abs(n))
	t_3 = sqrt(Float64(abs(U) * 2.0))
	tmp = 0.0
	if (l_m <= 1.02e-127)
		tmp = Float64(t_2 * Float64(t_3 * sqrt(abs(Float64(t - Float64(2.0 * t_1))))));
	elseif (l_m <= 4.4e+166)
		tmp = Float64(t_2 * Float64(t_3 * sqrt(abs(Float64(t - Float64(t_1 * Float64(2.0 + Float64(n * Float64(Float64(U - U_42_) / Om)))))))));
	else
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / Om) * sqrt(Float64(U * Float64(n * Float64(Float64(Om * -2.0) + Float64(n * U_42_))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (l_m ^ 2.0) / Om;
	t_2 = sqrt(abs(n));
	t_3 = sqrt((abs(U) * 2.0));
	tmp = 0.0;
	if (l_m <= 1.02e-127)
		tmp = t_2 * (t_3 * sqrt(abs((t - (2.0 * t_1)))));
	elseif (l_m <= 4.4e+166)
		tmp = t_2 * (t_3 * sqrt(abs((t - (t_1 * (2.0 + (n * ((U - U_42_) / Om))))))));
	else
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Abs[n], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Abs[U], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l$95$m, 1.02e-127], N[(t$95$2 * N[(t$95$3 * N[Sqrt[N[Abs[N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 4.4e+166], N[(t$95$2 * N[(t$95$3 * N[Sqrt[N[Abs[N[(t - N[(t$95$1 * N[(2.0 + N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(Om * -2.0), $MachinePrecision] + N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.02000000000000008e-127

   1. Initial program 54.4%

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

   3. Simplified 56.4%

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

   6. Applied egg-rr 35.3%

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

      1. unpow1/2 35.3%

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

      2. unpow2 35.3%

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

      3. rem-sqrt-square 53.7%

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

      4. associate-*r* 53.7%

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

      5. *-commutative 53.7%

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

      6. associate-*r/ 53.7%

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

   8. Simplified 53.7%

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

      1. pow1/2 53.7%

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

      2. fabs-mul 53.7%

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

      3. unpow-prod-down 63.0%

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

      4. associate-*l* 63.0%

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

   10. Applied egg-rr 63.0%

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

       1. unpow1/2 63.0%

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

       2. unpow1/2 63.0%

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

       3. associate-*r* 63.0%

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

   12. Simplified 63.0%

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

       1. pow1/2 63.0%

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

       2. fabs-mul 63.0%

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

       3. metadata-eval 63.0%

          \[\leadsto \sqrt{\left|n\right|} \cdot {\left(\left|U \cdot 2\right| \cdot \left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. unpow-prod-down 73.8%

          \[\leadsto \sqrt{\left|n\right|} \cdot \color{blue}{\left({\left(\left|U \cdot 2\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       5. metadata-eval 73.8%

          \[\leadsto \sqrt{\left|n\right|} \cdot \left({\left(\left|U \cdot 2\right|\right)}^{\color{blue}{0.5}} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right) \]

       6. metadata-eval 73.8%

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

   14. Applied egg-rr 73.8%

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

       1. unpow1/2 73.8%

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

       2. fabs-mul 73.8%

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

       3. metadata-eval 73.8%

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

       4. unpow1/2 73.8%

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

       5. fabs-sub 73.8%

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

   16. Simplified 73.8%

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

   17. Taylor expanded in n around 0 66.8%

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

   if 1.02000000000000008e-127 < l < 4.3999999999999998e166

   1. Initial program 44.0%

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

   3. Simplified 46.0%

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

   6. Applied egg-rr 33.8%

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

      1. unpow1/2 33.8%

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

      2. unpow2 33.8%

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

      3. rem-sqrt-square 50.4%

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

      4. associate-*r* 50.4%

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

      5. *-commutative 50.4%

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

      6. associate-*r/ 50.4%

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

   8. Simplified 50.4%

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

      1. pow1/2 50.4%

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

      2. fabs-mul 50.4%

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

      3. unpow-prod-down 62.4%

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

      4. associate-*l* 62.4%

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

   10. Applied egg-rr 62.4%

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

       1. unpow1/2 62.4%

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

       2. unpow1/2 62.4%

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

       3. associate-*r* 62.4%

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

   12. Simplified 62.4%

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

       1. pow1/2 62.4%

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

       2. fabs-mul 62.4%

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

       3. metadata-eval 62.4%

          \[\leadsto \sqrt{\left|n\right|} \cdot {\left(\left|U \cdot 2\right| \cdot \left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. unpow-prod-down 76.6%

          \[\leadsto \sqrt{\left|n\right|} \cdot \color{blue}{\left({\left(\left|U \cdot 2\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       5. metadata-eval 76.6%

          \[\leadsto \sqrt{\left|n\right|} \cdot \left({\left(\left|U \cdot 2\right|\right)}^{\color{blue}{0.5}} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right) \]

       6. metadata-eval 76.6%

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

   14. Applied egg-rr 76.6%

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

       1. unpow1/2 76.6%

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

       2. fabs-mul 76.6%

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

       3. metadata-eval 76.6%

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

       4. unpow1/2 76.6%

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

       5. fabs-sub 76.6%

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

   16. Simplified 76.6%

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

   17. Taylor expanded in U around -inf 61.5%

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

   19. Simplified 83.3%

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

   if 4.3999999999999998e166 < l

   1. Initial program 6.7%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in Om around 0 5.0%

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

   6. Taylor expanded in l around inf 41.0%

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

   7. Taylor expanded in U around 0 45.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - 2 \cdot \frac{{\ell}^{2}}{Om}\right|}\right)\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - \frac{{\ell}^{2}}{Om} \cdot \left(2 + n \cdot \frac{U - U*}{Om}\right)\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \frac{{l\_m}^{2}}{Om}\\ t_2 := \sqrt{\left|n\right|}\\ t_3 := t\_2 \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - t\_1\right|}\right)\\ \mathbf{if}\;l\_m \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;l\_m \leq 2.65 \cdot 10^{+76}:\\ \;\;\;\;t\_2 \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(\left(t\_1 + \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right)\right) - t\right)\right|}\\ \mathbf{elif}\;l\_m \leq 1.2 \cdot 10^{+165}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (pow l_m 2.0) Om)))
        (t_2 (sqrt (fabs n)))
        (t_3 (* t_2 (* (sqrt (* (fabs U) 2.0)) (sqrt (fabs (- t t_1)))))))
   (if (<= l_m 1.7e-78)
     t_3
     (if (<= l_m 2.65e+76)
       (*
        t_2
        (sqrt
         (fabs
          (*
           (* U 2.0)
           (- (+ t_1 (* (- U U*) (* n (pow (/ l_m Om) 2.0)))) t)))))
       (if (<= l_m 1.2e+165)
         t_3
         (*
          (/ (* l_m (sqrt 2.0)) Om)
          (sqrt (* U (* n (+ (* Om -2.0) (* n U*)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (pow(l_m, 2.0) / Om);
	double t_2 = sqrt(fabs(n));
	double t_3 = t_2 * (sqrt((fabs(U) * 2.0)) * sqrt(fabs((t - t_1))));
	double tmp;
	if (l_m <= 1.7e-78) {
		tmp = t_3;
	} else if (l_m <= 2.65e+76) {
		tmp = t_2 * sqrt(fabs(((U * 2.0) * ((t_1 + ((U - U_42_) * (n * pow((l_m / Om), 2.0)))) - t))));
	} else if (l_m <= 1.2e+165) {
		tmp = t_3;
	} else {
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 * ((l_m ** 2.0d0) / om)
    t_2 = sqrt(abs(n))
    t_3 = t_2 * (sqrt((abs(u) * 2.0d0)) * sqrt(abs((t - t_1))))
    if (l_m <= 1.7d-78) then
        tmp = t_3
    else if (l_m <= 2.65d+76) then
        tmp = t_2 * sqrt(abs(((u * 2.0d0) * ((t_1 + ((u - u_42) * (n * ((l_m / om) ** 2.0d0)))) - t))))
    else if (l_m <= 1.2d+165) then
        tmp = t_3
    else
        tmp = ((l_m * sqrt(2.0d0)) / om) * sqrt((u * (n * ((om * (-2.0d0)) + (n * u_42)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (Math.pow(l_m, 2.0) / Om);
	double t_2 = Math.sqrt(Math.abs(n));
	double t_3 = t_2 * (Math.sqrt((Math.abs(U) * 2.0)) * Math.sqrt(Math.abs((t - t_1))));
	double tmp;
	if (l_m <= 1.7e-78) {
		tmp = t_3;
	} else if (l_m <= 2.65e+76) {
		tmp = t_2 * Math.sqrt(Math.abs(((U * 2.0) * ((t_1 + ((U - U_42_) * (n * Math.pow((l_m / Om), 2.0)))) - t))));
	} else if (l_m <= 1.2e+165) {
		tmp = t_3;
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) / Om) * Math.sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (math.pow(l_m, 2.0) / Om)
	t_2 = math.sqrt(math.fabs(n))
	t_3 = t_2 * (math.sqrt((math.fabs(U) * 2.0)) * math.sqrt(math.fabs((t - t_1))))
	tmp = 0
	if l_m <= 1.7e-78:
		tmp = t_3
	elif l_m <= 2.65e+76:
		tmp = t_2 * math.sqrt(math.fabs(((U * 2.0) * ((t_1 + ((U - U_42_) * (n * math.pow((l_m / Om), 2.0)))) - t))))
	elif l_m <= 1.2e+165:
		tmp = t_3
	else:
		tmp = ((l_m * math.sqrt(2.0)) / Om) * math.sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64((l_m ^ 2.0) / Om))
	t_2 = sqrt(abs(n))
	t_3 = Float64(t_2 * Float64(sqrt(Float64(abs(U) * 2.0)) * sqrt(abs(Float64(t - t_1)))))
	tmp = 0.0
	if (l_m <= 1.7e-78)
		tmp = t_3;
	elseif (l_m <= 2.65e+76)
		tmp = Float64(t_2 * sqrt(abs(Float64(Float64(U * 2.0) * Float64(Float64(t_1 + Float64(Float64(U - U_42_) * Float64(n * (Float64(l_m / Om) ^ 2.0)))) - t)))));
	elseif (l_m <= 1.2e+165)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / Om) * sqrt(Float64(U * Float64(n * Float64(Float64(Om * -2.0) + Float64(n * U_42_))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * ((l_m ^ 2.0) / Om);
	t_2 = sqrt(abs(n));
	t_3 = t_2 * (sqrt((abs(U) * 2.0)) * sqrt(abs((t - t_1))));
	tmp = 0.0;
	if (l_m <= 1.7e-78)
		tmp = t_3;
	elseif (l_m <= 2.65e+76)
		tmp = t_2 * sqrt(abs(((U * 2.0) * ((t_1 + ((U - U_42_) * (n * ((l_m / Om) ^ 2.0)))) - t))));
	elseif (l_m <= 1.2e+165)
		tmp = t_3;
	else
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Abs[n], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[Sqrt[N[(N[Abs[U], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(t - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 1.7e-78], t$95$3, If[LessEqual[l$95$m, 2.65e+76], N[(t$95$2 * N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(N[(t$95$1 + N[(N[(U - U$42$), $MachinePrecision] * N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.2e+165], t$95$3, N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(Om * -2.0), $MachinePrecision] + N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.70000000000000006e-78 or 2.65000000000000008e76 < l < 1.2e165

   1. Initial program 52.8%

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

   3. Simplified 54.5%

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

   6. Applied egg-rr 33.7%

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

      1. unpow1/2 33.7%

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

      2. unpow2 33.7%

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

      3. rem-sqrt-square 52.6%

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

      4. associate-*r* 52.7%

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

      5. *-commutative 52.7%

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

      6. associate-*r/ 52.7%

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

   8. Simplified 52.7%

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

      1. pow1/2 52.7%

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

      2. fabs-mul 52.7%

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

      3. unpow-prod-down 61.5%

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

      4. associate-*l* 61.5%

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

   10. Applied egg-rr 61.5%

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

       1. unpow1/2 61.5%

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

       2. unpow1/2 61.5%

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

       3. associate-*r* 61.5%

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

   12. Simplified 61.5%

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

       1. pow1/2 61.5%

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

       2. fabs-mul 61.5%

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

       3. metadata-eval 61.5%

          \[\leadsto \sqrt{\left|n\right|} \cdot {\left(\left|U \cdot 2\right| \cdot \left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. unpow-prod-down 73.0%

          \[\leadsto \sqrt{\left|n\right|} \cdot \color{blue}{\left({\left(\left|U \cdot 2\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       5. metadata-eval 73.0%

          \[\leadsto \sqrt{\left|n\right|} \cdot \left({\left(\left|U \cdot 2\right|\right)}^{\color{blue}{0.5}} \cdot {\left(\left|t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right|\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right) \]

       6. metadata-eval 73.0%

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

   14. Applied egg-rr 73.0%

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

       1. unpow1/2 73.0%

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

       2. fabs-mul 73.0%

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

       3. metadata-eval 73.0%

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

       4. unpow1/2 73.0%

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

       5. fabs-sub 73.0%

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

   16. Simplified 73.0%

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

   17. Taylor expanded in n around 0 68.5%

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

   if 1.70000000000000006e-78 < l < 2.65000000000000008e76

   1. Initial program 49.2%

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

   3. Simplified 52.7%

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

   6. Applied egg-rr 45.6%

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

      1. unpow1/2 45.6%

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

      2. unpow2 45.6%

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

      3. rem-sqrt-square 56.5%

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

      4. associate-*r* 56.5%

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

      5. *-commutative 56.5%

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

      6. associate-*r/ 56.5%

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

   8. Simplified 56.5%

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

      1. pow1/2 56.5%

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

      2. fabs-mul 56.5%

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

      3. unpow-prod-down 74.2%

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

      4. associate-*l* 74.2%

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

   10. Applied egg-rr 74.2%

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

       1. unpow1/2 74.2%

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

       2. unpow1/2 74.2%

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

       3. associate-*r* 74.2%

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

   12. Simplified 74.2%

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

       1. fma-undefine 74.2%

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

   14. Applied egg-rr 74.2%

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

   if 1.2e165 < l

   1. Initial program 6.7%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in Om around 0 5.0%

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

   6. Taylor expanded in l around inf 41.0%

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

   7. Taylor expanded in U around 0 45.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - 2 \cdot \frac{{\ell}^{2}}{Om}\right|}\right)\\ \mathbf{elif}\;\ell \leq 2.65 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \sqrt{\left|\left(U \cdot 2\right) \cdot \left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) - t\right)\right|}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\left|n\right|} \cdot \left(\sqrt{\left|U\right| \cdot 2} \cdot \sqrt{\left|t - 2 \cdot \frac{{\ell}^{2}}{Om}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt(((U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(Math.abs(n)) * Math.sqrt(Math.abs((2.0 * (U * t))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt(((U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(math.fabs(n)) * math.sqrt(math.fabs((2.0 * (U * t))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt(((U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(abs(n)) * sqrt(abs((2.0 * (U * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = ((l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.9%

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

   3. Simplified 51.4%

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

   6. Applied egg-rr 30.9%

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

      1. unpow1/2 30.9%

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

      2. unpow2 30.9%

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

      3. rem-sqrt-square 54.1%

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

      4. associate-*r* 54.1%

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

      5. *-commutative 54.1%

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

      6. associate-*r/ 54.1%

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

   8. Simplified 54.1%

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

      1. pow1/2 54.1%

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

      2. fabs-mul 54.1%

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

      3. unpow-prod-down 82.5%

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

      4. associate-*l* 82.5%

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

   10. Applied egg-rr 82.5%

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

       1. unpow1/2 82.5%

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

       2. unpow1/2 82.5%

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

       3. associate-*r* 82.5%

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

   12. Simplified 82.5%

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

   13. Taylor expanded in t around inf 72.6%

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

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

   1. Initial program 68.8%

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

   3. Simplified 73.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 10.7%

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

   5. Taylor expanded in Om around 0 8.9%

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

   6. Taylor expanded in l around inf 32.9%

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

      1. associate-*l/ 31.3%

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

      2. fma-define 31.3%

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

      3. mul-1-neg 31.3%

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

      4. fmm-undef 31.3%

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

      5. *-commutative 31.3%

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

      6. associate-*r* 31.3%

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

   8. Applied egg-rr 31.3%

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

4. Final simplification 65.5%

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

Alternative 7: 65.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(Math.abs(n)) * Math.sqrt(Math.abs(((U * 2.0) * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(math.fabs(n)) * math.sqrt(math.fabs(((U * 2.0) * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(abs(n)) * sqrt(abs(((U * 2.0) * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = ((l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 43.5%

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

   6. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

      2. unpow2 26.9%

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

      3. rem-sqrt-square 46.4%

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

      4. associate-*r* 46.4%

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

      5. *-commutative 46.4%

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

      6. associate-*r/ 46.4%

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

   8. Simplified 46.4%

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

      1. pow1/2 46.4%

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

      2. fabs-mul 46.4%

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

      3. unpow-prod-down 70.2%

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

      4. associate-*l* 70.2%

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

   10. Applied egg-rr 70.2%

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

       1. unpow1/2 70.2%

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

       2. unpow1/2 70.2%

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

       3. associate-*r* 70.2%

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

   12. Simplified 70.2%

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

   13. Taylor expanded in n around 0 74.6%

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

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

   1. Initial program 68.8%

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

   3. Simplified 73.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in Om around 0 5.4%

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

   6. Taylor expanded in l around inf 35.9%

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

      1. associate-*l/ 34.1%

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

      2. fma-define 34.1%

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

      3. mul-1-neg 34.1%

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

      4. fmm-undef 34.1%

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

      5. *-commutative 34.1%

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

      6. associate-*r* 34.1%

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

   8. Applied egg-rr 34.1%

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

4. Final simplification 67.4%

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

Alternative 8: 60.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(Math.abs((n * ((U * 2.0) * ((2.0 * (Math.pow(l_m, 2.0) / Om)) - t)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(math.fabs((n * ((U * 2.0) * ((2.0 * (math.pow(l_m, 2.0) / Om)) - t)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(abs((n * ((U * 2.0) * ((2.0 * ((l_m ^ 2.0) / Om)) - t)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = ((l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 43.5%

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

   6. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

      2. unpow2 26.9%

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

      3. rem-sqrt-square 46.4%

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

      4. associate-*r* 46.4%

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

      5. *-commutative 46.4%

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

      6. associate-*r/ 46.4%

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

   8. Simplified 46.4%

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

   9. Taylor expanded in n around 0 50.8%

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

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

   1. Initial program 68.8%

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

   3. Simplified 73.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in Om around 0 5.4%

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

   6. Taylor expanded in l around inf 35.9%

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

      1. associate-*l/ 34.1%

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

      2. fma-define 34.1%

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

      3. mul-1-neg 34.1%

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

      4. fmm-undef 34.1%

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

      5. *-commutative 34.1%

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

      6. associate-*r* 34.1%

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

   8. Applied egg-rr 34.1%

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

4. Final simplification 63.3%

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

Alternative 9: 60.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) * Math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) * math.sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (U * (n * 2.0)) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = ((l_m * sqrt(2.0)) * sqrt((n * ((-2.0 * (U * Om)) + ((U_42_ - U) * (n * U)))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in n around 0 50.0%

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

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

   1. Initial program 68.8%

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

   3. Simplified 73.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in Om around 0 5.4%

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

   6. Taylor expanded in l around inf 35.9%

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

      1. associate-*l/ 34.1%

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

      2. fma-define 34.1%

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

      3. mul-1-neg 34.1%

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

      4. fmm-undef 34.1%

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

      5. *-commutative 34.1%

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

      6. associate-*r* 34.1%

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

   8. Applied egg-rr 34.1%

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

4. Final simplification 63.2%

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

Alternative 10: 43.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+215}:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t + \frac{{l\_m}^{2} \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -2.7e+215)
   (pow (* t (* U (* n 2.0))) 0.5)
   (if (<= n 6e+42)
     (sqrt (* (* U 2.0) (* n (+ t (/ (* (pow l_m 2.0) -2.0) Om)))))
     (* (sqrt (* n 2.0)) (sqrt (* U t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -2.7e+215) {
		tmp = pow((t * (U * (n * 2.0))), 0.5);
	} else if (n <= 6e+42) {
		tmp = sqrt(((U * 2.0) * (n * (t + ((pow(l_m, 2.0) * -2.0) / Om)))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-2.7d+215)) then
        tmp = (t * (u * (n * 2.0d0))) ** 0.5d0
    else if (n <= 6d+42) then
        tmp = sqrt(((u * 2.0d0) * (n * (t + (((l_m ** 2.0d0) * (-2.0d0)) / om)))))
    else
        tmp = sqrt((n * 2.0d0)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -2.7e+215) {
		tmp = Math.pow((t * (U * (n * 2.0))), 0.5);
	} else if (n <= 6e+42) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t + ((Math.pow(l_m, 2.0) * -2.0) / Om)))));
	} else {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -2.7e+215:
		tmp = math.pow((t * (U * (n * 2.0))), 0.5)
	elif n <= 6e+42:
		tmp = math.sqrt(((U * 2.0) * (n * (t + ((math.pow(l_m, 2.0) * -2.0) / Om)))))
	else:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -2.7e+215)
		tmp = Float64(t * Float64(U * Float64(n * 2.0))) ^ 0.5;
	elseif (n <= 6e+42)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t + Float64(Float64((l_m ^ 2.0) * -2.0) / Om)))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= -2.7e+215)
		tmp = (t * (U * (n * 2.0))) ^ 0.5;
	elseif (n <= 6e+42)
		tmp = sqrt(((U * 2.0) * (n * (t + (((l_m ^ 2.0) * -2.0) / Om)))));
	else
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -2.7e+215], N[Power[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 6e+42], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.7e215

   1. Initial program 39.6%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in Om around 0 34.2%

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

   6. Taylor expanded in t around inf 33.9%

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

      1. pow1/2 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-*r* 45.9%

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

   8. Applied egg-rr 45.9%

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

   if -2.7e215 < n < 6.00000000000000058e42

   1. Initial program 49.5%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in n around 0 51.6%

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

      1. associate-*r* 51.6%

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

      2. *-commutative 51.6%

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

      3. cancel-sign-sub-inv 51.6%

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

      4. metadata-eval 51.6%

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

      5. associate-*r/ 51.6%

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

      6. *-commutative 51.6%

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

   7. Simplified 51.6%

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

   if 6.00000000000000058e42 < n

   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 51.0%

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

   5. Taylor expanded in t around inf 44.3%

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

      1. sqrt-prod 58.3%

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

      2. *-commutative 58.3%

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

   7. Applied egg-rr 58.3%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+215}:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -0.00034:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 760000000000:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -0.00034)
   (pow (* t (* U (* n 2.0))) 0.5)
   (if (<= U 760000000000.0)
     (sqrt (* (* n 2.0) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (sqrt (fabs (* (* U 2.0) (* n t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -0.00034) {
		tmp = pow((t * (U * (n * 2.0))), 0.5);
	} else if (U <= 760000000000.0) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else {
		tmp = sqrt(fabs(((U * 2.0) * (n * t))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-0.00034d0)) then
        tmp = (t * (u * (n * 2.0d0))) ** 0.5d0
    else if (u <= 760000000000.0d0) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else
        tmp = sqrt(abs(((u * 2.0d0) * (n * t))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -0.00034) {
		tmp = Math.pow((t * (U * (n * 2.0))), 0.5);
	} else if (U <= 760000000000.0) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else {
		tmp = Math.sqrt(Math.abs(((U * 2.0) * (n * t))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -0.00034:
		tmp = math.pow((t * (U * (n * 2.0))), 0.5)
	elif U <= 760000000000.0:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	else:
		tmp = math.sqrt(math.fabs(((U * 2.0) * (n * t))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -0.00034)
		tmp = Float64(t * Float64(U * Float64(n * 2.0))) ^ 0.5;
	elseif (U <= 760000000000.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	else
		tmp = sqrt(abs(Float64(Float64(U * 2.0) * Float64(n * t))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -0.00034)
		tmp = (t * (U * (n * 2.0))) ^ 0.5;
	elseif (U <= 760000000000.0)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	else
		tmp = sqrt(abs(((U * 2.0) * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -0.00034], N[Power[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 760000000000.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -3.4e-4

   1. Initial program 54.6%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in Om around 0 55.7%

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

   6. Taylor expanded in t around inf 49.2%

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

      1. pow1/2 55.0%

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

      2. *-commutative 55.0%

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

      3. associate-*r* 55.0%

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

   8. Applied egg-rr 55.0%

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

   if -3.4e-4 < U < 7.6e11

   1. Initial program 47.5%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in n around 0 50.5%

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

   if 7.6e11 < U

   1. Initial program 46.8%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around inf 38.8%

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

      1. add-sqr-sqrt 38.8%

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

      2. pow1/2 38.8%

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

      3. pow1/2 42.7%

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

      4. pow-prod-down 27.0%

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

      5. pow2 27.0%

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

      6. associate-*r* 27.0%

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

      7. *-commutative 27.0%

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

   7. Applied egg-rr 27.0%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}^{2}\right)}^{0.5}}} \]
   8. Step-by-step derivation

      1. unpow1/2 27.0%

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

      2. unpow2 27.0%

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

      3. rem-sqrt-square 43.9%

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

   9. Simplified 43.9%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -0.00034:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 760000000000:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.8 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t + \frac{{l\_m}^{2} \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.8e+159)
   (sqrt (* (* U 2.0) (* n (+ t (/ (* (pow l_m 2.0) -2.0) Om)))))
   (* (/ (* l_m (sqrt 2.0)) Om) (sqrt (* U (* n (+ (* Om -2.0) (* n U*))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.8e+159) {
		tmp = sqrt(((U * 2.0) * (n * (t + ((pow(l_m, 2.0) * -2.0) / Om)))));
	} else {
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.8d+159) then
        tmp = sqrt(((u * 2.0d0) * (n * (t + (((l_m ** 2.0d0) * (-2.0d0)) / om)))))
    else
        tmp = ((l_m * sqrt(2.0d0)) / om) * sqrt((u * (n * ((om * (-2.0d0)) + (n * u_42)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.8e+159) {
		tmp = Math.sqrt(((U * 2.0) * (n * (t + ((Math.pow(l_m, 2.0) * -2.0) / Om)))));
	} else {
		tmp = ((l_m * Math.sqrt(2.0)) / Om) * Math.sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.8e+159:
		tmp = math.sqrt(((U * 2.0) * (n * (t + ((math.pow(l_m, 2.0) * -2.0) / Om)))))
	else:
		tmp = ((l_m * math.sqrt(2.0)) / Om) * math.sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.8e+159)
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t + Float64(Float64((l_m ^ 2.0) * -2.0) / Om)))));
	else
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / Om) * sqrt(Float64(U * Float64(n * Float64(Float64(Om * -2.0) + Float64(n * U_42_))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.8e+159)
		tmp = sqrt(((U * 2.0) * (n * (t + (((l_m ^ 2.0) * -2.0) / Om)))));
	else
		tmp = ((l_m * sqrt(2.0)) / Om) * sqrt((U * (n * ((Om * -2.0) + (n * U_42_)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.8e+159], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(Om * -2.0), $MachinePrecision] + N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.8000000000000001e159

   1. Initial program 52.4%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in n around 0 50.4%

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

      1. associate-*r* 50.4%

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

      2. *-commutative 50.4%

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

      3. cancel-sign-sub-inv 50.4%

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

      4. metadata-eval 50.4%

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

      5. associate-*r/ 50.4%

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

      6. *-commutative 50.4%

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

   7. Simplified 50.4%

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

   if 2.8000000000000001e159 < l

   1. Initial program 6.7%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in Om around 0 5.0%

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

   6. Taylor expanded in l around inf 41.0%

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

   7. Taylor expanded in U around 0 45.6%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(n \cdot \left(Om \cdot -2 + n \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\left|\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 6.8e-97)
   (sqrt (fabs (* (* U 2.0) (* n t))))
   (* (sqrt (* n 2.0)) (sqrt (* U t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 6.8e-97) {
		tmp = sqrt(fabs(((U * 2.0) * (n * t))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 6.8d-97) then
        tmp = sqrt(abs(((u * 2.0d0) * (n * t))))
    else
        tmp = sqrt((n * 2.0d0)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 6.8e-97) {
		tmp = Math.sqrt(Math.abs(((U * 2.0) * (n * t))));
	} else {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 6.8e-97:
		tmp = math.sqrt(math.fabs(((U * 2.0) * (n * t))))
	else:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 6.8e-97)
		tmp = sqrt(abs(Float64(Float64(U * 2.0) * Float64(n * t))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 6.8e-97)
		tmp = sqrt(abs(((U * 2.0) * (n * t))));
	else
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 6.8e-97], N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.7999999999999998e-97

   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 50.2%

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

   5. Taylor expanded in t around inf 41.5%

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

      1. add-sqr-sqrt 41.5%

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

      2. pow1/2 41.5%

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

      3. pow1/2 42.8%

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

      4. pow-prod-down 29.0%

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

      5. pow2 29.0%

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

      6. associate-*r* 29.0%

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

      7. *-commutative 29.0%

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

   7. Applied egg-rr 29.0%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}^{2}\right)}^{0.5}}} \]
   8. Step-by-step derivation

      1. unpow1/2 29.0%

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

      2. unpow2 29.0%

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

      3. rem-sqrt-square 43.6%

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

   9. Simplified 43.6%

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

   if 6.7999999999999998e-97 < n

   1. Initial program 55.4%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in t around inf 45.3%

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

      1. sqrt-prod 54.1%

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

      2. *-commutative 54.1%

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

   7. Applied egg-rr 54.1%

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

Alternative 14: 38.2% accurate, 1.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (fabs (* (* U 2.0) (* n t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[Abs[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.8%

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

3. Simplified 52.5%

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

5. Taylor expanded in t around inf 41.8%

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

   1. add-sqr-sqrt 41.8%

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

   2. pow1/2 41.8%

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

   3. pow1/2 43.5%

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

   4. pow-prod-down 30.0%

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

   5. pow2 30.0%

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

   6. associate-*r* 30.0%

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

   7. *-commutative 30.0%

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

7. Applied egg-rr 30.0%

   \[\leadsto \sqrt{\color{blue}{{\left({\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}^{2}\right)}^{0.5}}} \]
8. Step-by-step derivation

   1. unpow1/2 30.0%

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

   2. unpow2 30.0%

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

   3. rem-sqrt-square 44.3%

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

9. Simplified 44.3%

   \[\leadsto \sqrt{\color{blue}{\left|\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right|}} \]
10. Add Preprocessing

Alternative 15: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -2.7 \cdot 10^{-138}:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -2.7e-138)
   (pow (* t (* U (* n 2.0))) 0.5)
   (pow (* (* U 2.0) (* n t)) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2.7e-138) {
		tmp = pow((t * (U * (n * 2.0))), 0.5);
	} else {
		tmp = pow(((U * 2.0) * (n * t)), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-2.7d-138)) then
        tmp = (t * (u * (n * 2.0d0))) ** 0.5d0
    else
        tmp = ((u * 2.0d0) * (n * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2.7e-138) {
		tmp = Math.pow((t * (U * (n * 2.0))), 0.5);
	} else {
		tmp = Math.pow(((U * 2.0) * (n * t)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -2.7e-138:
		tmp = math.pow((t * (U * (n * 2.0))), 0.5)
	else:
		tmp = math.pow(((U * 2.0) * (n * t)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -2.7e-138)
		tmp = Float64(t * Float64(U * Float64(n * 2.0))) ^ 0.5;
	else
		tmp = Float64(Float64(U * 2.0) * Float64(n * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -2.7e-138)
		tmp = (t * (U * (n * 2.0))) ^ 0.5;
	else
		tmp = ((U * 2.0) * (n * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -2.7e-138], N[Power[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -2.70000000000000029e-138

   1. Initial program 51.4%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in Om around 0 50.8%

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

   6. Taylor expanded in t around inf 44.9%

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

      1. pow1/2 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*r* 48.4%

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

   8. Applied egg-rr 48.4%

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

   if -2.70000000000000029e-138 < U

   1. Initial program 47.5%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in t around inf 41.7%

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

      1. pow1/2 43.6%

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

      2. associate-*r* 43.6%

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

      3. *-commutative 43.6%

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

   7. Applied egg-rr 43.6%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.7 \cdot 10^{-138}:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -2.4 \cdot 10^{-138}:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -2.4e-138)
   (pow (* t (* U (* n 2.0))) 0.5)
   (sqrt (* 2.0 (* U (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2.4e-138) {
		tmp = pow((t * (U * (n * 2.0))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-2.4d-138)) then
        tmp = (t * (u * (n * 2.0d0))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2.4e-138) {
		tmp = Math.pow((t * (U * (n * 2.0))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -2.4e-138:
		tmp = math.pow((t * (U * (n * 2.0))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -2.4e-138)
		tmp = Float64(t * Float64(U * Float64(n * 2.0))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (U <= -2.4e-138)
		tmp = (t * (U * (n * 2.0))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -2.4e-138], N[Power[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -2.3999999999999999e-138

   1. Initial program 51.4%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in Om around 0 50.8%

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

   6. Taylor expanded in t around inf 44.9%

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

      1. pow1/2 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*r* 48.4%

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

   8. Applied egg-rr 48.4%

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

   if -2.3999999999999999e-138 < U

   1. Initial program 47.5%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in t around inf 41.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.4 \cdot 10^{-138}:\\ \;\;\;\;{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 2.4e+38)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 2.4e+38) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 2.4d+38) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 2.4e+38) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= 2.4e+38:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 2.4e+38)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= 2.4e+38)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, 2.4e+38], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 2.40000000000000017e38

   1. Initial program 48.1%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in t around inf 41.1%

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

   if 2.40000000000000017e38 < Om

   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 67.3%

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

   5. Taylor expanded in t around inf 51.1%

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

      1. pow1/2 53.0%

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

      2. associate-*l* 53.0%

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

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.4% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.8%

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

3. Simplified 52.5%

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

5. Taylor expanded in t around inf 41.8%

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

Reproduce

herbie shell --seed 2024166 
(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*))))))