Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 65.1%

Time: 29.1s

Alternatives: 26

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 5e-156)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
     (if (<= t_1 4e+151)
       t_1
       (sqrt
        (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 5e-156) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U))))))));
	} else if (t_1 <= 4e+151) {
		tmp = t_1;
	} else {
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 5e-156)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	elseif (t_1 <= 4e+151)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-156], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+151], t$95$1, N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000007e-156

   1. Initial program 18.1%

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

   3. Simplified 39.4%

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

      1. sqrt-prod 46.6%

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

   5. Applied egg-rr 46.6%

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

      1. *-commutative 46.6%

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

      2. *-commutative 46.6%

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

   7. Simplified 46.6%

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

   if 5.00000000000000007e-156 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 4.00000000000000007e151

   1. Initial program 99.8%

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

   if 4.00000000000000007e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 20.1%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in t around 0 37.0%

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

   5. Taylor expanded in U around 0 38.0%

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

   6. Taylor expanded in l around 0 40.5%

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

      1. associate-*r* 39.7%

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

      2. sub-neg 39.7%

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

      3. associate-/l* 38.2%

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

      4. metadata-eval 38.2%

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

   8. Simplified 38.2%

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

      1. associate-*r/ 40.8%

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

      2. *-commutative 40.8%

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

      3. +-commutative 40.8%

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

      4. associate-/r/ 43.1%

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

   10. Applied egg-rr 43.1%

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

       1. associate-*r* 49.9%

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

       2. *-commutative 49.9%

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

       3. +-commutative 49.9%

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

       4. associate-*l/ 49.9%

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

       5. metadata-eval 49.9%

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

       6. sub-neg 49.9%

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

       7. associate-*r* 52.2%

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

       8. *-commutative 52.2%

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

       9. sub-neg 52.2%

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

       10. associate-*l/ 52.2%

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

       11. metadata-eval 52.2%

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

       12. +-commutative 52.2%

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

       13. *-commutative 52.2%

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

   12. Simplified 52.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 4 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 2: 64.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 0.0)
     (sqrt
      (+ (* 2.0 (* n (* U t))) (* 2.0 (/ (* -2.0 (* n (* U (* l l)))) Om))))
     (if (<= t_1 5e+302)
       (sqrt t_1)
       (sqrt
        (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((-2.0 * (n * (U * (l * l)))) / Om))));
	} else if (t_1 <= 5e+302) {
		tmp = sqrt(t_1);
	} else {
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * (((-2.0d0) * (n * (u * (l * l)))) / om))))
    else if (t_1 <= 5d+302) then
        tmp = sqrt(t_1)
    else
        tmp = sqrt(((((2.0d0 * n) * l) * (l * (u * ((-2.0d0) + (u_42 * (n / om)))))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((-2.0 * (n * (U * (l * l)))) / Om))));
	} else if (t_1 <= 5e+302) {
		tmp = Math.sqrt(t_1);
	} else {
		tmp = Math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((-2.0 * (n * (U * (l * l)))) / Om))))
	elif t_1 <= 5e+302:
		tmp = math.sqrt(t_1)
	else:
		tmp = math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(-2.0 * Float64(n * Float64(U * Float64(l * l)))) / Om))));
	elseif (t_1 <= 5e+302)
		tmp = sqrt(t_1);
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((-2.0 * (n * (U * (l * l)))) / Om))));
	elseif (t_1 <= 5e+302)
		tmp = sqrt(t_1);
	else
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(-2.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[Sqrt[t$95$1], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.0%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t around inf 37.3%

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

   5. Taylor expanded in n around 0 39.1%

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

      1. *-commutative 39.1%

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

      2. unpow2 39.1%

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

   7. Simplified 39.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 5e302

   1. Initial program 99.3%

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

   if 5e302 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 20.2%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in t around 0 37.2%

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

   5. Taylor expanded in U around 0 38.3%

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

   6. Taylor expanded in l around 0 40.8%

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

      1. associate-*r* 40.0%

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

      2. sub-neg 40.0%

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

      3. associate-/l* 38.5%

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

      4. metadata-eval 38.5%

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

   8. Simplified 38.5%

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

      1. associate-*r/ 40.4%

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

      2. *-commutative 40.4%

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

      3. +-commutative 40.4%

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

      4. associate-/r/ 42.7%

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

   10. Applied egg-rr 42.7%

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

       1. associate-*r* 49.5%

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

       2. *-commutative 49.5%

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

       3. +-commutative 49.5%

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

       4. associate-*l/ 49.5%

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

       5. metadata-eval 49.5%

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

       6. sub-neg 49.5%

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

       7. associate-*r* 51.8%

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

       8. *-commutative 51.8%

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

       9. sub-neg 51.8%

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

       10. associate-*l/ 51.8%

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

       11. metadata-eval 51.8%

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

       12. +-commutative 51.8%

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

       13. *-commutative 51.8%

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

   12. Simplified 51.8%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 3: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-244}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{t_2}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot t_2}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0))) (t_2 (* n (* U l))))
   (if (<= l -1.35e+104)
     (* t_1 (- (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om))))
     (if (<= l -2.8e-244)
       (pow
        (*
         2.0
         (+
          (* U (* n t))
          (/ (+ (/ (* n (* l (- U* U))) Om) (* l -2.0)) (/ Om t_2))))
        0.5)
       (if (<= l 2.02e+73)
         (sqrt
          (*
           (* 2.0 n)
           (*
            U
            (-
             t
             (+
              (* 2.0 (/ l (/ Om l)))
              (* n (* (pow (/ l Om) 2.0) (- U U*))))))))
         (if (<= l 6e+192)
           (sqrt
            (+
             (* 2.0 (* n (* U t)))
             (*
              2.0
              (/ (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) t_2) Om))))
           (*
            t_1
            (sqrt (/ (* n (* U (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double t_2 = n * (U * l);
	double tmp;
	if (l <= -1.35e+104) {
		tmp = t_1 * -sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	} else if (l <= -2.8e-244) {
		tmp = pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5);
	} else if (l <= 2.02e+73) {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l / (Om / l))) + (n * (pow((l / Om), 2.0) * (U - U_42_))))))));
	} else if (l <= 6e+192) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	} else {
		tmp = t_1 * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l * sqrt(2.0d0)
    t_2 = n * (u * l)
    if (l <= (-1.35d+104)) then
        tmp = t_1 * -sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / om))
    else if (l <= (-2.8d-244)) then
        tmp = (2.0d0 * ((u * (n * t)) + ((((n * (l * (u_42 - u))) / om) + (l * (-2.0d0))) / (om / t_2)))) ** 0.5d0
    else if (l <= 2.02d+73) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - ((2.0d0 * (l / (om / l))) + (n * (((l / om) ** 2.0d0) * (u - u_42))))))))
    else if (l <= 6d+192) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * t_2) / om))))
    else
        tmp = t_1 * sqrt(((n * (u * (((n * (u_42 - u)) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * Math.sqrt(2.0);
	double t_2 = n * (U * l);
	double tmp;
	if (l <= -1.35e+104) {
		tmp = t_1 * -Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	} else if (l <= -2.8e-244) {
		tmp = Math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5);
	} else if (l <= 2.02e+73) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - ((2.0 * (l / (Om / l))) + (n * (Math.pow((l / Om), 2.0) * (U - U_42_))))))));
	} else if (l <= 6e+192) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	} else {
		tmp = t_1 * Math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * math.sqrt(2.0)
	t_2 = n * (U * l)
	tmp = 0
	if l <= -1.35e+104:
		tmp = t_1 * -math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	elif l <= -2.8e-244:
		tmp = math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5)
	elif l <= 2.02e+73:
		tmp = math.sqrt(((2.0 * n) * (U * (t - ((2.0 * (l / (Om / l))) + (n * (math.pow((l / Om), 2.0) * (U - U_42_))))))))
	elif l <= 6e+192:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))))
	else:
		tmp = t_1 * math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	t_2 = Float64(n * Float64(U * l))
	tmp = 0.0
	if (l <= -1.35e+104)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om))));
	elseif (l <= -2.8e-244)
		tmp = Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(Float64(Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om) + Float64(l * -2.0)) / Float64(Om / t_2)))) ^ 0.5;
	elseif (l <= 2.02e+73)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * Float64(l / Float64(Om / l))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_))))))));
	elseif (l <= 6e+192)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * t_2) / Om))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * sqrt(2.0);
	t_2 = n * (U * l);
	tmp = 0.0;
	if (l <= -1.35e+104)
		tmp = t_1 * -sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	elseif (l <= -2.8e-244)
		tmp = (2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))) ^ 0.5;
	elseif (l <= 2.02e+73)
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l / (Om / l))) + (n * (((l / Om) ^ 2.0) * (U - U_42_))))))));
	elseif (l <= 6e+192)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	else
		tmp = t_1 * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.35e+104], N[(t$95$1 * (-N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2.8e-244], N[Power[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.02e+73], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+192], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.34999999999999992e104

   1. Initial program 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 36.8%

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

   5. Taylor expanded in U around 0 37.3%

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

   6. Taylor expanded in l around 0 43.6%

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

      1. associate-*r* 43.6%

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

      2. sub-neg 43.6%

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

      3. associate-/l* 43.6%

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

      4. metadata-eval 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in l around -inf 70.1%

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

   if -1.34999999999999992e104 < l < -2.80000000000000013e-244

   1. Initial program 53.5%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 47.0%

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

   5. Taylor expanded in U* around 0 47.0%

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

      1. associate-*r* 47.0%

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

      2. mul-1-neg 47.0%

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

      3. associate-*r* 47.0%

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

      4. distribute-rgt-neg-in 47.0%

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

      5. distribute-lft-in 47.0%

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

      6. sub-neg 47.0%

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

   7. Simplified 47.0%

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

      1. pow1/2 47.0%

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

      2. distribute-lft-out 47.0%

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

      3. associate-*r* 54.2%

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

      4. associate-/l* 55.5%

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

      5. associate-*l* 55.6%

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

      6. *-commutative 55.6%

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

      7. *-commutative 55.6%

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

   9. Applied egg-rr 55.6%

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

   if -2.80000000000000013e-244 < l < 2.01999999999999994e73

   1. Initial program 72.6%

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

      1. associate-*l* 69.5%

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

      2. sub-neg 69.5%

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

      3. associate-+l- 69.5%

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

      4. sub-neg 69.5%

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

      5. associate-/l* 69.5%

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

      6. remove-double-neg 69.5%

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

      7. associate-*l* 69.5%

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

   3. Simplified 69.5%

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

   if 2.01999999999999994e73 < l < 6e192

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 6e192 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 36.2%

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

   5. Taylor expanded in l around inf 93.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-244}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 4: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+108}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.75 \cdot 10^{-246}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{t_1}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot t_1}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{\frac{Om}{U}}{-2 + \frac{n}{\frac{Om}{U* - U}}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U l))))
   (if (<= l -4.5e+108)
     (* (* l (sqrt 2.0)) (- (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om))))
     (if (<= l -1.75e-246)
       (pow
        (*
         2.0
         (+
          (* U (* n t))
          (/ (+ (/ (* n (* l (- U* U))) Om) (* l -2.0)) (/ Om t_1))))
        0.5)
       (if (<= l 3.9e+74)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
         (if (<= l 4.4e+193)
           (sqrt
            (+
             (* 2.0 (* n (* U t)))
             (*
              2.0
              (/ (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) t_1) Om))))
           (*
            (sqrt 2.0)
            (*
             l
             (sqrt (/ n (/ (/ Om U) (+ -2.0 (/ n (/ Om (- U* U)))))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double tmp;
	if (l <= -4.5e+108) {
		tmp = (l * sqrt(2.0)) * -sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	} else if (l <= -1.75e-246) {
		tmp = pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))), 0.5);
	} else if (l <= 3.9e+74) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 4.4e+193) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / ((Om / U) / (-2.0 + (n / (Om / (U_42_ - U))))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (u * l)
    if (l <= (-4.5d+108)) then
        tmp = (l * sqrt(2.0d0)) * -sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / om))
    else if (l <= (-1.75d-246)) then
        tmp = (2.0d0 * ((u * (n * t)) + ((((n * (l * (u_42 - u))) / om) + (l * (-2.0d0))) / (om / t_1)))) ** 0.5d0
    else if (l <= 3.9d+74) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 4.4d+193) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * t_1) / om))))
    else
        tmp = sqrt(2.0d0) * (l * sqrt((n / ((om / u) / ((-2.0d0) + (n / (om / (u_42 - u))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double tmp;
	if (l <= -4.5e+108) {
		tmp = (l * Math.sqrt(2.0)) * -Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	} else if (l <= -1.75e-246) {
		tmp = Math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))), 0.5);
	} else if (l <= 3.9e+74) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 4.4e+193) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((n / ((Om / U) / (-2.0 + (n / (Om / (U_42_ - U))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U * l)
	tmp = 0
	if l <= -4.5e+108:
		tmp = (l * math.sqrt(2.0)) * -math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	elif l <= -1.75e-246:
		tmp = math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))), 0.5)
	elif l <= 3.9e+74:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 4.4e+193:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((n / ((Om / U) / (-2.0 + (n / (Om / (U_42_ - U))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	tmp = 0.0
	if (l <= -4.5e+108)
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(-sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om))));
	elseif (l <= -1.75e-246)
		tmp = Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(Float64(Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om) + Float64(l * -2.0)) / Float64(Om / t_1)))) ^ 0.5;
	elseif (l <= 3.9e+74)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 4.4e+193)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * t_1) / Om))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Float64(Om / U) / Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U * l);
	tmp = 0.0;
	if (l <= -4.5e+108)
		tmp = (l * sqrt(2.0)) * -sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	elseif (l <= -1.75e-246)
		tmp = (2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))) ^ 0.5;
	elseif (l <= 3.9e+74)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 4.4e+193)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))));
	else
		tmp = sqrt(2.0) * (l * sqrt((n / ((Om / U) / (-2.0 + (n / (Om / (U_42_ - U))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+108], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -1.75e-246], N[Power[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 3.9e+74], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.4e+193], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(N[(Om / U), $MachinePrecision] / N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -4.5e108

   1. Initial program 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 36.8%

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

   5. Taylor expanded in U around 0 37.3%

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

   6. Taylor expanded in l around 0 43.6%

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

      1. associate-*r* 43.6%

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

      2. sub-neg 43.6%

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

      3. associate-/l* 43.6%

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

      4. metadata-eval 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in l around -inf 70.1%

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

   if -4.5e108 < l < -1.7500000000000001e-246

   1. Initial program 53.5%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 47.0%

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

   5. Taylor expanded in U* around 0 47.0%

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

      1. associate-*r* 47.0%

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

      2. mul-1-neg 47.0%

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

      3. associate-*r* 47.0%

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

      4. distribute-rgt-neg-in 47.0%

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

      5. distribute-lft-in 47.0%

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

      6. sub-neg 47.0%

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

   7. Simplified 47.0%

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

      1. pow1/2 47.0%

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

      2. distribute-lft-out 47.0%

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

      3. associate-*r* 54.2%

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

      4. associate-/l* 55.5%

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

      5. associate-*l* 55.6%

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

      6. *-commutative 55.6%

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

      7. *-commutative 55.6%

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

   9. Applied egg-rr 55.6%

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

   if -1.7500000000000001e-246 < l < 3.90000000000000008e74

   1. Initial program 72.6%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in U around 0 69.5%

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

   if 3.90000000000000008e74 < l < 4.39999999999999972e193

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 4.39999999999999972e193 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 36.2%

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

   5. Taylor expanded in l around inf 93.9%

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

      1. associate-*l* 93.8%

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

      2. associate-/l* 93.8%

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

      3. *-commutative 93.8%

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

      4. associate-/r* 93.9%

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

      5. sub-neg 93.9%

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

      6. associate-/l* 91.8%

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

      7. metadata-eval 91.8%

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

   7. Simplified 91.8%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{\frac{Om}{U}}{\frac{n}{\frac{Om}{U* - U}} + -2}}}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+108}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.75 \cdot 10^{-246}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{\frac{Om}{U}}{-2 + \frac{n}{\frac{Om}{U* - U}}}}}\right)\\ \end{array} \]

Alternative 5: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+104}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{-244}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{t_2}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot t_2}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0))) (t_2 (* n (* U l))))
   (if (<= l -5.2e+104)
     (* t_1 (- (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om))))
     (if (<= l -1.02e-244)
       (pow
        (*
         2.0
         (+
          (* U (* n t))
          (/ (+ (/ (* n (* l (- U* U))) Om) (* l -2.0)) (/ Om t_2))))
        0.5)
       (if (<= l 4.6e+74)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
         (if (<= l 6e+192)
           (sqrt
            (+
             (* 2.0 (* n (* U t)))
             (*
              2.0
              (/ (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) t_2) Om))))
           (*
            t_1
            (sqrt (/ (* n (* U (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double t_2 = n * (U * l);
	double tmp;
	if (l <= -5.2e+104) {
		tmp = t_1 * -sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	} else if (l <= -1.02e-244) {
		tmp = pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5);
	} else if (l <= 4.6e+74) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 6e+192) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	} else {
		tmp = t_1 * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l * sqrt(2.0d0)
    t_2 = n * (u * l)
    if (l <= (-5.2d+104)) then
        tmp = t_1 * -sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / om))
    else if (l <= (-1.02d-244)) then
        tmp = (2.0d0 * ((u * (n * t)) + ((((n * (l * (u_42 - u))) / om) + (l * (-2.0d0))) / (om / t_2)))) ** 0.5d0
    else if (l <= 4.6d+74) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 6d+192) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * t_2) / om))))
    else
        tmp = t_1 * sqrt(((n * (u * (((n * (u_42 - u)) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * Math.sqrt(2.0);
	double t_2 = n * (U * l);
	double tmp;
	if (l <= -5.2e+104) {
		tmp = t_1 * -Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	} else if (l <= -1.02e-244) {
		tmp = Math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5);
	} else if (l <= 4.6e+74) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 6e+192) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	} else {
		tmp = t_1 * Math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * math.sqrt(2.0)
	t_2 = n * (U * l)
	tmp = 0
	if l <= -5.2e+104:
		tmp = t_1 * -math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	elif l <= -1.02e-244:
		tmp = math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5)
	elif l <= 4.6e+74:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 6e+192:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))))
	else:
		tmp = t_1 * math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	t_2 = Float64(n * Float64(U * l))
	tmp = 0.0
	if (l <= -5.2e+104)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om))));
	elseif (l <= -1.02e-244)
		tmp = Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(Float64(Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om) + Float64(l * -2.0)) / Float64(Om / t_2)))) ^ 0.5;
	elseif (l <= 4.6e+74)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 6e+192)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * t_2) / Om))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * sqrt(2.0);
	t_2 = n * (U * l);
	tmp = 0.0;
	if (l <= -5.2e+104)
		tmp = t_1 * -sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	elseif (l <= -1.02e-244)
		tmp = (2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))) ^ 0.5;
	elseif (l <= 4.6e+74)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 6e+192)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	else
		tmp = t_1 * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.2e+104], N[(t$95$1 * (-N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -1.02e-244], N[Power[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 4.6e+74], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+192], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -5.20000000000000001e104

   1. Initial program 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 36.8%

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

   5. Taylor expanded in U around 0 37.3%

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

   6. Taylor expanded in l around 0 43.6%

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

      1. associate-*r* 43.6%

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

      2. sub-neg 43.6%

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

      3. associate-/l* 43.6%

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

      4. metadata-eval 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in l around -inf 70.1%

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

   if -5.20000000000000001e104 < l < -1.02000000000000006e-244

   1. Initial program 53.5%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 47.0%

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

   5. Taylor expanded in U* around 0 47.0%

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

      1. associate-*r* 47.0%

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

      2. mul-1-neg 47.0%

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

      3. associate-*r* 47.0%

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

      4. distribute-rgt-neg-in 47.0%

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

      5. distribute-lft-in 47.0%

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

      6. sub-neg 47.0%

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

   7. Simplified 47.0%

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

      1. pow1/2 47.0%

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

      2. distribute-lft-out 47.0%

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

      3. associate-*r* 54.2%

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

      4. associate-/l* 55.5%

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

      5. associate-*l* 55.6%

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

      6. *-commutative 55.6%

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

      7. *-commutative 55.6%

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

   9. Applied egg-rr 55.6%

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

   if -1.02000000000000006e-244 < l < 4.5999999999999997e74

   1. Initial program 72.6%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in U around 0 69.5%

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

   if 4.5999999999999997e74 < l < 6e192

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 6e192 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 36.2%

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

   5. Taylor expanded in l around inf 93.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right) \cdot U\right)}{Om}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+104}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -1.02 \cdot 10^{-244}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 6: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\right)\\ \mathbf{elif}\;\ell \leq -8.8 \cdot 10^{-245}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{t_1}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot t_1}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U l))))
   (if (<= l -1.65e+106)
     (*
      (sqrt 2.0)
      (* l (- (sqrt (/ n (/ Om (* U (+ -2.0 (* U* (/ n Om))))))))))
     (if (<= l -8.8e-245)
       (pow
        (*
         2.0
         (+
          (* U (* n t))
          (/ (+ (/ (* n (* l (- U* U))) Om) (* l -2.0)) (/ Om t_1))))
        0.5)
       (if (<= l 1.3e+77)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
         (if (<= l 2.6e+198)
           (sqrt
            (+
             (* 2.0 (* n (* U t)))
             (*
              2.0
              (/ (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) t_1) Om))))
           (*
            (* l (sqrt 2.0))
            (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double tmp;
	if (l <= -1.65e+106) {
		tmp = sqrt(2.0) * (l * -sqrt((n / (Om / (U * (-2.0 + (U_42_ * (n / Om))))))));
	} else if (l <= -8.8e-245) {
		tmp = pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))), 0.5);
	} else if (l <= 1.3e+77) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 2.6e+198) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (u * l)
    if (l <= (-1.65d+106)) then
        tmp = sqrt(2.0d0) * (l * -sqrt((n / (om / (u * ((-2.0d0) + (u_42 * (n / om))))))))
    else if (l <= (-8.8d-245)) then
        tmp = (2.0d0 * ((u * (n * t)) + ((((n * (l * (u_42 - u))) / om) + (l * (-2.0d0))) / (om / t_1)))) ** 0.5d0
    else if (l <= 1.3d+77) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 2.6d+198) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * t_1) / om))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double tmp;
	if (l <= -1.65e+106) {
		tmp = Math.sqrt(2.0) * (l * -Math.sqrt((n / (Om / (U * (-2.0 + (U_42_ * (n / Om))))))));
	} else if (l <= -8.8e-245) {
		tmp = Math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))), 0.5);
	} else if (l <= 1.3e+77) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 2.6e+198) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U * l)
	tmp = 0
	if l <= -1.65e+106:
		tmp = math.sqrt(2.0) * (l * -math.sqrt((n / (Om / (U * (-2.0 + (U_42_ * (n / Om))))))))
	elif l <= -8.8e-245:
		tmp = math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))), 0.5)
	elif l <= 1.3e+77:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 2.6e+198:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	tmp = 0.0
	if (l <= -1.65e+106)
		tmp = Float64(sqrt(2.0) * Float64(l * Float64(-sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om))))))))));
	elseif (l <= -8.8e-245)
		tmp = Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(Float64(Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om) + Float64(l * -2.0)) / Float64(Om / t_1)))) ^ 0.5;
	elseif (l <= 1.3e+77)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 2.6e+198)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * t_1) / Om))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U * l);
	tmp = 0.0;
	if (l <= -1.65e+106)
		tmp = sqrt(2.0) * (l * -sqrt((n / (Om / (U * (-2.0 + (U_42_ * (n / Om))))))));
	elseif (l <= -8.8e-245)
		tmp = (2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_1)))) ^ 0.5;
	elseif (l <= 1.3e+77)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 2.6e+198)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_1) / Om))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.65e+106], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * (-N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -8.8e-245], N[Power[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.3e+77], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.6e+198], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.65000000000000004e106

   1. Initial program 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 36.8%

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

   5. Taylor expanded in U around 0 37.3%

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

   6. Taylor expanded in l around 0 43.6%

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

      1. associate-*r* 43.6%

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

      2. sub-neg 43.6%

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

      3. associate-/l* 43.6%

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

      4. metadata-eval 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in l around -inf 70.1%

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

       1. mul-1-neg 70.1%

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

       2. associate-*l* 70.0%

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

       3. associate-/l* 68.6%

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

       4. *-commutative 68.6%

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

       5. sub-neg 68.6%

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

       6. associate-*l/ 68.6%

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

       7. metadata-eval 68.6%

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

       8. +-commutative 68.6%

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

       9. *-commutative 68.6%

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

   11. Simplified 68.6%

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

   if -1.65000000000000004e106 < l < -8.79999999999999971e-245

   1. Initial program 53.5%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 47.0%

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

   5. Taylor expanded in U* around 0 47.0%

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

      1. associate-*r* 47.0%

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

      2. mul-1-neg 47.0%

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

      3. associate-*r* 47.0%

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

      4. distribute-rgt-neg-in 47.0%

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

      5. distribute-lft-in 47.0%

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

      6. sub-neg 47.0%

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

   7. Simplified 47.0%

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

      1. pow1/2 47.0%

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

      2. distribute-lft-out 47.0%

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

      3. associate-*r* 54.2%

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

      4. associate-/l* 55.5%

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

      5. associate-*l* 55.6%

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

      6. *-commutative 55.6%

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

      7. *-commutative 55.6%

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

   9. Applied egg-rr 55.6%

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

   if -8.79999999999999971e-245 < l < 1.3000000000000001e77

   1. Initial program 72.6%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in U around 0 69.5%

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

   if 1.3000000000000001e77 < l < 2.59999999999999981e198

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 2.59999999999999981e198 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around 0 31.2%

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

   5. Taylor expanded in U around 0 31.9%

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

   6. Taylor expanded in l around 0 89.6%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\right)\\ \mathbf{elif}\;\ell \leq -8.8 \cdot 10^{-245}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 7: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := n \cdot \left(U \cdot \ell\right)\\ t_3 := \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+107}:\\ \;\;\;\;t_1 \cdot \left(-t_3\right)\\ \mathbf{elif}\;\ell \leq -6.6 \cdot 10^{-244}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{t_2}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot t_2}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0)))
        (t_2 (* n (* U l)))
        (t_3 (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om))))
   (if (<= l -1.95e+107)
     (* t_1 (- t_3))
     (if (<= l -6.6e-244)
       (pow
        (*
         2.0
         (+
          (* U (* n t))
          (/ (+ (/ (* n (* l (- U* U))) Om) (* l -2.0)) (/ Om t_2))))
        0.5)
       (if (<= l 4e+76)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
         (if (<= l 6e+192)
           (sqrt
            (+
             (* 2.0 (* n (* U t)))
             (*
              2.0
              (/ (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) t_2) Om))))
           (* t_1 t_3)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double t_2 = n * (U * l);
	double t_3 = sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	double tmp;
	if (l <= -1.95e+107) {
		tmp = t_1 * -t_3;
	} else if (l <= -6.6e-244) {
		tmp = pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5);
	} else if (l <= 4e+76) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 6e+192) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	} else {
		tmp = t_1 * t_3;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = l * sqrt(2.0d0)
    t_2 = n * (u * l)
    t_3 = sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / om))
    if (l <= (-1.95d+107)) then
        tmp = t_1 * -t_3
    else if (l <= (-6.6d-244)) then
        tmp = (2.0d0 * ((u * (n * t)) + ((((n * (l * (u_42 - u))) / om) + (l * (-2.0d0))) / (om / t_2)))) ** 0.5d0
    else if (l <= 4d+76) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 6d+192) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * t_2) / om))))
    else
        tmp = t_1 * t_3
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * Math.sqrt(2.0);
	double t_2 = n * (U * l);
	double t_3 = Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	double tmp;
	if (l <= -1.95e+107) {
		tmp = t_1 * -t_3;
	} else if (l <= -6.6e-244) {
		tmp = Math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5);
	} else if (l <= 4e+76) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 6e+192) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	} else {
		tmp = t_1 * t_3;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * math.sqrt(2.0)
	t_2 = n * (U * l)
	t_3 = math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	tmp = 0
	if l <= -1.95e+107:
		tmp = t_1 * -t_3
	elif l <= -6.6e-244:
		tmp = math.pow((2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))), 0.5)
	elif l <= 4e+76:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 6e+192:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))))
	else:
		tmp = t_1 * t_3
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	t_2 = Float64(n * Float64(U * l))
	t_3 = sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om))
	tmp = 0.0
	if (l <= -1.95e+107)
		tmp = Float64(t_1 * Float64(-t_3));
	elseif (l <= -6.6e-244)
		tmp = Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(Float64(Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om) + Float64(l * -2.0)) / Float64(Om / t_2)))) ^ 0.5;
	elseif (l <= 4e+76)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 6e+192)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * t_2) / Om))));
	else
		tmp = Float64(t_1 * t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * sqrt(2.0);
	t_2 = n * (U * l);
	t_3 = sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	tmp = 0.0;
	if (l <= -1.95e+107)
		tmp = t_1 * -t_3;
	elseif (l <= -6.6e-244)
		tmp = (2.0 * ((U * (n * t)) + ((((n * (l * (U_42_ - U))) / Om) + (l * -2.0)) / (Om / t_2)))) ^ 0.5;
	elseif (l <= 4e+76)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 6e+192)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * t_2) / Om))));
	else
		tmp = t_1 * t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.95e+107], N[(t$95$1 * (-t$95$3)), $MachinePrecision], If[LessEqual[l, -6.6e-244], N[Power[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 4e+76], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+192], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.9499999999999999e107

   1. Initial program 26.9%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 36.8%

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

   5. Taylor expanded in U around 0 37.3%

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

   6. Taylor expanded in l around 0 43.6%

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

      1. associate-*r* 43.6%

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

      2. sub-neg 43.6%

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

      3. associate-/l* 43.6%

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

      4. metadata-eval 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in l around -inf 70.1%

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

   if -1.9499999999999999e107 < l < -6.60000000000000052e-244

   1. Initial program 53.5%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around inf 47.0%

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

   5. Taylor expanded in U* around 0 47.0%

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

      1. associate-*r* 47.0%

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

      2. mul-1-neg 47.0%

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

      3. associate-*r* 47.0%

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

      4. distribute-rgt-neg-in 47.0%

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

      5. distribute-lft-in 47.0%

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

      6. sub-neg 47.0%

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

   7. Simplified 47.0%

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

      1. pow1/2 47.0%

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

      2. distribute-lft-out 47.0%

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

      3. associate-*r* 54.2%

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

      4. associate-/l* 55.5%

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

      5. associate-*l* 55.6%

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

      6. *-commutative 55.6%

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

      7. *-commutative 55.6%

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

   9. Applied egg-rr 55.6%

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

   if -6.60000000000000052e-244 < l < 4.0000000000000002e76

   1. Initial program 72.6%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in U around 0 69.5%

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

   if 4.0000000000000002e76 < l < 6e192

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 6e192 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around 0 31.2%

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

   5. Taylor expanded in U around 0 31.9%

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

   6. Taylor expanded in l around 0 89.6%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{n \cdot \left(\left(\frac{n \cdot U*}{Om} - 2\right) \cdot U\right)}{Om}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+107}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\right)\\ \mathbf{elif}\;\ell \leq -6.6 \cdot 10^{-244}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right) + \frac{\frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 8: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot t_1\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 2.22 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{t_1}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (+ -2.0 (* U* (/ n Om))))))
   (if (<= l -9.5e-18)
     (sqrt (/ (* (* (* 2.0 n) l) (* l t_1)) Om))
     (if (<= l 2.22e+77)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
       (if (<= l 5.5e+192)
         (sqrt
          (+
           (* 2.0 (* n (* U t)))
           (*
            2.0
            (/
             (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) (* n (* U l)))
             Om))))
         (* (sqrt 2.0) (* l (sqrt (/ n (/ Om t_1))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (-2.0 + (U_42_ * (n / Om)));
	double tmp;
	if (l <= -9.5e-18) {
		tmp = sqrt(((((2.0 * n) * l) * (l * t_1)) / Om));
	} else if (l <= 2.22e+77) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 5.5e+192) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * ((-2.0d0) + (u_42 * (n / om)))
    if (l <= (-9.5d-18)) then
        tmp = sqrt(((((2.0d0 * n) * l) * (l * t_1)) / om))
    else if (l <= 2.22d+77) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 5.5d+192) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * (n * (u * l))) / om))))
    else
        tmp = sqrt(2.0d0) * (l * sqrt((n / (om / t_1))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (-2.0 + (U_42_ * (n / Om)));
	double tmp;
	if (l <= -9.5e-18) {
		tmp = Math.sqrt(((((2.0 * n) * l) * (l * t_1)) / Om));
	} else if (l <= 2.22e+77) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 5.5e+192) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((n / (Om / t_1))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U * (-2.0 + (U_42_ * (n / Om)))
	tmp = 0
	if l <= -9.5e-18:
		tmp = math.sqrt(((((2.0 * n) * l) * (l * t_1)) / Om))
	elif l <= 2.22e+77:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 5.5e+192:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((n / (Om / t_1))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om))))
	tmp = 0.0
	if (l <= -9.5e-18)
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * t_1)) / Om));
	elseif (l <= 2.22e+77)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 5.5e+192)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * Float64(n * Float64(U * l))) / Om))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (-2.0 + (U_42_ * (n / Om)));
	tmp = 0.0;
	if (l <= -9.5e-18)
		tmp = sqrt(((((2.0 * n) * l) * (l * t_1)) / Om));
	elseif (l <= 2.22e+77)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 5.5e+192)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	else
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9.5e-18], N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.22e+77], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.5e+192], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.5000000000000003e-18

   1. Initial program 36.3%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in t around 0 40.6%

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

   5. Taylor expanded in U around 0 41.0%

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

   6. Taylor expanded in l around 0 45.7%

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

      1. associate-*r* 45.7%

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

      2. sub-neg 45.7%

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

      3. associate-/l* 44.2%

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

      4. metadata-eval 44.2%

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

   8. Simplified 44.2%

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

      1. associate-*r/ 45.7%

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

      2. *-commutative 45.7%

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

      3. +-commutative 45.7%

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

      4. associate-/r/ 48.7%

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

   10. Applied egg-rr 48.7%

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

       1. associate-*r* 58.2%

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

       2. *-commutative 58.2%

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

       3. +-commutative 58.2%

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

       4. associate-*l/ 58.2%

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

       5. metadata-eval 58.2%

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

       6. sub-neg 58.2%

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

       7. associate-*r* 59.7%

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

       8. *-commutative 59.7%

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

       9. sub-neg 59.7%

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

       10. associate-*l/ 59.7%

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

       11. metadata-eval 59.7%

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

       12. +-commutative 59.7%

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

       13. *-commutative 59.7%

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

   12. Simplified 59.7%

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

   if -9.5000000000000003e-18 < l < 2.22e77

   1. Initial program 64.0%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in U around 0 60.4%

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

   if 2.22e77 < l < 5.49999999999999966e192

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 5.49999999999999966e192 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around inf 36.2%

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

      1. *-un-lft-identity 36.2%

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

      2. distribute-lft-out 36.2%

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

      3. associate-/l* 41.4%

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

      4. associate-/l* 41.4%

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

      5. *-commutative 41.4%

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

      6. *-commutative 41.4%

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

   6. Applied egg-rr 41.4%

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

      1. *-lft-identity 41.4%

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

      2. fma-def 41.4%

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

      3. *-commutative 41.4%

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

      4. associate-/l* 41.4%

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

      5. +-commutative 41.4%

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

      6. fma-def 41.4%

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

      7. associate-*l/ 41.4%

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

      8. *-commutative 41.4%

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

      9. associate-/r* 41.6%

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

      10. associate-/r* 41.3%

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

   8. Simplified 41.3%

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

   9. Taylor expanded in U* around inf 42.0%

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

   10. Taylor expanded in l around inf 89.6%

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

       1. associate-*l* 89.5%

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

       2. associate-/l* 89.5%

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

       3. *-commutative 89.5%

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

       4. sub-neg 89.5%

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

       5. associate-*l/ 89.5%

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

       6. metadata-eval 89.5%

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

       7. +-commutative 89.5%

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

       8. *-commutative 89.5%

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

   12. Simplified 89.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 2.22 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 9: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -9.5e-18)
   (sqrt (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))
   (if (<= l 8.8e+75)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (if (<= l 5.5e+192)
       (sqrt
        (+
         (* 2.0 (* n (* U t)))
         (*
          2.0
          (/
           (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) (* n (* U l)))
           Om))))
       (*
        (* l (sqrt 2.0))
        (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -9.5e-18) {
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	} else if (l <= 8.8e+75) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 5.5e+192) {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-9.5d-18)) then
        tmp = sqrt(((((2.0d0 * n) * l) * (l * (u * ((-2.0d0) + (u_42 * (n / om)))))) / om))
    else if (l <= 8.8d+75) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 5.5d+192) then
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * (n * (u * l))) / om))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -9.5e-18) {
		tmp = Math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	} else if (l <= 8.8e+75) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 5.5e+192) {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -9.5e-18:
		tmp = math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om))
	elif l <= 8.8e+75:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 5.5e+192:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -9.5e-18)
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om));
	elseif (l <= 8.8e+75)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 5.5e+192)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * Float64(n * Float64(U * l))) / Om))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -9.5e-18)
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	elseif (l <= 8.8e+75)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 5.5e+192)
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -9.5e-18], N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.8e+75], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.5e+192], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.5000000000000003e-18

   1. Initial program 36.3%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in t around 0 40.6%

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

   5. Taylor expanded in U around 0 41.0%

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

   6. Taylor expanded in l around 0 45.7%

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

      1. associate-*r* 45.7%

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

      2. sub-neg 45.7%

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

      3. associate-/l* 44.2%

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

      4. metadata-eval 44.2%

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

   8. Simplified 44.2%

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

      1. associate-*r/ 45.7%

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

      2. *-commutative 45.7%

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

      3. +-commutative 45.7%

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

      4. associate-/r/ 48.7%

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

   10. Applied egg-rr 48.7%

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

       1. associate-*r* 58.2%

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

       2. *-commutative 58.2%

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

       3. +-commutative 58.2%

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

       4. associate-*l/ 58.2%

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

       5. metadata-eval 58.2%

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

       6. sub-neg 58.2%

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

       7. associate-*r* 59.7%

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

       8. *-commutative 59.7%

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

       9. sub-neg 59.7%

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

       10. associate-*l/ 59.7%

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

       11. metadata-eval 59.7%

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

       12. +-commutative 59.7%

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

       13. *-commutative 59.7%

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

   12. Simplified 59.7%

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

   if -9.5000000000000003e-18 < l < 8.80000000000000048e75

   1. Initial program 64.0%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in U around 0 60.4%

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

   if 8.80000000000000048e75 < l < 5.49999999999999966e192

   1. Initial program 31.1%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in t around inf 76.2%

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

   5. Taylor expanded in U* around 0 72.1%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. associate-*r* 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

      5. distribute-lft-in 88.4%

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

      6. sub-neg 88.4%

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

   7. Simplified 88.4%

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

   if 5.49999999999999966e192 < l

   1. Initial program 1.3%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in t around 0 31.2%

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

   5. Taylor expanded in U around 0 31.9%

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

   6. Taylor expanded in l around 0 89.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 10: 58.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -9.5e-18)
   (sqrt (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))
   (if (<= l 6e+75)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (sqrt
      (+
       (* 2.0 (* n (* U t)))
       (*
        2.0
        (/
         (* (+ (/ (* (- U* U) (* n l)) Om) (* l -2.0)) (* n (* U l)))
         Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -9.5e-18) {
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	} else if (l <= 6e+75) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-9.5d-18)) then
        tmp = sqrt(((((2.0d0 * n) * l) * (l * (u * ((-2.0d0) + (u_42 * (n / om)))))) / om))
    else if (l <= 6d+75) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt(((2.0d0 * (n * (u * t))) + (2.0d0 * ((((((u_42 - u) * (n * l)) / om) + (l * (-2.0d0))) * (n * (u * l))) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -9.5e-18) {
		tmp = Math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	} else if (l <= 6e+75) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -9.5e-18:
		tmp = math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om))
	elif l <= 6e+75:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -9.5e-18)
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om));
	elseif (l <= 6e+75)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) + Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om) + Float64(l * -2.0)) * Float64(n * Float64(U * l))) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -9.5e-18)
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	elseif (l <= 6e+75)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(((2.0 * (n * (U * t))) + (2.0 * ((((((U_42_ - U) * (n * l)) / Om) + (l * -2.0)) * (n * (U * l))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -9.5e-18], N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+75], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.5000000000000003e-18

   1. Initial program 36.3%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in t around 0 40.6%

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

   5. Taylor expanded in U around 0 41.0%

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

   6. Taylor expanded in l around 0 45.7%

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

      1. associate-*r* 45.7%

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

      2. sub-neg 45.7%

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

      3. associate-/l* 44.2%

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

      4. metadata-eval 44.2%

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

   8. Simplified 44.2%

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

      1. associate-*r/ 45.7%

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

      2. *-commutative 45.7%

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

      3. +-commutative 45.7%

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

      4. associate-/r/ 48.7%

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

   10. Applied egg-rr 48.7%

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

       1. associate-*r* 58.2%

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

       2. *-commutative 58.2%

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

       3. +-commutative 58.2%

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

       4. associate-*l/ 58.2%

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

       5. metadata-eval 58.2%

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

       6. sub-neg 58.2%

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

       7. associate-*r* 59.7%

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

       8. *-commutative 59.7%

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

       9. sub-neg 59.7%

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

       10. associate-*l/ 59.7%

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

       11. metadata-eval 59.7%

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

       12. +-commutative 59.7%

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

       13. *-commutative 59.7%

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

   12. Simplified 59.7%

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

   if -9.5000000000000003e-18 < l < 6e75

   1. Initial program 64.0%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in U around 0 60.4%

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

   if 6e75 < l

   1. Initial program 18.3%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in t around inf 59.1%

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

   5. Taylor expanded in U* around 0 54.3%

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

      1. associate-*r* 63.5%

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

      2. mul-1-neg 63.5%

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

      3. associate-*r* 56.5%

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

      4. distribute-rgt-neg-in 56.5%

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

      5. distribute-lft-in 68.5%

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

      6. sub-neg 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om} + \ell \cdot -2\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \end{array} \]

Alternative 11: 58.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18} \lor \neg \left(\ell \leq 8.2 \cdot 10^{+55}\right):\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -9.5e-18) (not (<= l 8.2e+55)))
   (sqrt (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -9.5e-18) || !(l <= 8.2e+55)) {
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((l <= (-9.5d-18)) .or. (.not. (l <= 8.2d+55))) then
        tmp = sqrt(((((2.0d0 * n) * l) * (l * (u * ((-2.0d0) + (u_42 * (n / om)))))) / om))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -9.5e-18) || !(l <= 8.2e+55)) {
		tmp = Math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -9.5e-18) or not (l <= 8.2e+55):
		tmp = math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -9.5e-18) || !(l <= 8.2e+55))
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -9.5e-18) || ~((l <= 8.2e+55)))
		tmp = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -9.5e-18], N[Not[LessEqual[l, 8.2e+55]], $MachinePrecision]], N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.5000000000000003e-18 or 8.19999999999999962e55 < l

   1. Initial program 31.9%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in t around 0 43.2%

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

   5. Taylor expanded in U around 0 43.5%

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

   6. Taylor expanded in l around 0 46.2%

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

      1. associate-*r* 46.2%

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

      2. sub-neg 46.2%

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

      3. associate-/l* 45.4%

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

      4. metadata-eval 45.4%

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

   8. Simplified 45.4%

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

      1. associate-*r/ 48.4%

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

      2. *-commutative 48.4%

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

      3. +-commutative 48.4%

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

      4. associate-/r/ 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. associate-*r* 60.5%

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

       2. *-commutative 60.5%

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

       3. +-commutative 60.5%

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

       4. associate-*l/ 60.5%

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

       5. metadata-eval 60.5%

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

       6. sub-neg 60.5%

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

       7. associate-*r* 62.2%

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

       8. *-commutative 62.2%

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

       9. sub-neg 62.2%

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

       10. associate-*l/ 62.2%

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

       11. metadata-eval 62.2%

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

       12. +-commutative 62.2%

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

       13. *-commutative 62.2%

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

   12. Simplified 62.2%

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

   if -9.5000000000000003e-18 < l < 8.19999999999999962e55

   1. Initial program 63.6%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in U around 0 60.5%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-18} \lor \neg \left(\ell \leq 8.2 \cdot 10^{+55}\right):\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 12: 52.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{-271}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (* 2.0 n) (* (/ l Om) (* l (* U (+ -2.0 (* U* (/ n Om))))))))))
   (if (<= l -3.8e-35)
     t_1
     (if (<= l -1.55e-271)
       (pow (* 2.0 (* U (* n t))) 0.5)
       (if (<= l 8.4e+49)
         (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))
         t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * ((l / Om) * (l * (U * (-2.0 + (U_42_ * (n / Om))))))));
	double tmp;
	if (l <= -3.8e-35) {
		tmp = t_1;
	} else if (l <= -1.55e-271) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 8.4e+49) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * ((l / om) * (l * (u * ((-2.0d0) + (u_42 * (n / om))))))))
    if (l <= (-3.8d-35)) then
        tmp = t_1
    else if (l <= (-1.55d-271)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 8.4d+49) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * n) * ((l / Om) * (l * (U * (-2.0 + (U_42_ * (n / Om))))))));
	double tmp;
	if (l <= -3.8e-35) {
		tmp = t_1;
	} else if (l <= -1.55e-271) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 8.4e+49) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * ((l / Om) * (l * (U * (-2.0 + (U_42_ * (n / Om))))))))
	tmp = 0
	if l <= -3.8e-35:
		tmp = t_1
	elif l <= -1.55e-271:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 8.4e+49:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(l / Om) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om))))))))
	tmp = 0.0
	if (l <= -3.8e-35)
		tmp = t_1;
	elseif (l <= -1.55e-271)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 8.4e+49)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * ((l / Om) * (l * (U * (-2.0 + (U_42_ * (n / Om))))))));
	tmp = 0.0;
	if (l <= -3.8e-35)
		tmp = t_1;
	elseif (l <= -1.55e-271)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 8.4e+49)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.8e-35], t$95$1, If[LessEqual[l, -1.55e-271], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 8.4e+49], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.8000000000000001e-35 or 8.40000000000000043e49 < l

   1. Initial program 32.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in t around 0 43.4%

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

   5. Taylor expanded in U around 0 43.7%

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

   6. Taylor expanded in l around 0 46.3%

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

      1. associate-*r* 46.4%

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

      2. sub-neg 46.4%

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

      3. associate-/l* 45.5%

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

      4. metadata-eval 45.5%

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

   8. Simplified 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. associate-*l* 45.5%

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

      3. associate-/l* 51.5%

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

      4. *-commutative 51.5%

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

      5. +-commutative 51.5%

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

      6. associate-/r/ 52.3%

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

   10. Applied egg-rr 52.3%

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

       1. *-lft-identity 52.3%

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

       2. associate-*r* 52.3%

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

       3. associate-/r/ 52.3%

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

       4. *-commutative 52.3%

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

       5. +-commutative 52.3%

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

       6. associate-*l/ 52.3%

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

       7. metadata-eval 52.3%

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

       8. sub-neg 52.3%

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

       9. associate-*r* 52.2%

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

       10. *-commutative 52.2%

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

       11. sub-neg 52.2%

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

       12. associate-*l/ 52.2%

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

       13. metadata-eval 52.2%

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

       14. +-commutative 52.2%

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

       15. *-commutative 52.2%

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

   12. Simplified 52.2%

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

   if -3.8000000000000001e-35 < l < -1.54999999999999995e-271

   1. Initial program 52.3%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around inf 32.7%

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

      1. *-un-lft-identity 32.7%

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

   6. Applied egg-rr 32.7%

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

      1. *-lft-identity 32.7%

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

      2. associate-*r* 41.8%

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

   8. Simplified 41.8%

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

      1. pow1/2 45.3%

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

      2. *-commutative 45.3%

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

   10. Applied egg-rr 45.3%

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

   if -1.54999999999999995e-271 < l < 8.40000000000000043e49

   1. Initial program 71.9%

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

   2. Taylor expanded in Om around inf 61.7%

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

      1. unpow2 61.7%

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

      2. associate-*r/ 61.7%

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

   4. Simplified 61.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{-271}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13: 52.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-270}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))))
   (if (<= l -3.2e-32)
     t_1
     (if (<= l -1e-270)
       (pow (* 2.0 (* U (* n t))) 0.5)
       (if (<= l 2.2e+43)
         (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))
         t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	double tmp;
	if (l <= -3.2e-32) {
		tmp = t_1;
	} else if (l <= -1e-270) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2.2e+43) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((((2.0d0 * n) * l) * (l * (u * ((-2.0d0) + (u_42 * (n / om)))))) / om))
    if (l <= (-3.2d-32)) then
        tmp = t_1
    else if (l <= (-1d-270)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 2.2d+43) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	double tmp;
	if (l <= -3.2e-32) {
		tmp = t_1;
	} else if (l <= -1e-270) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2.2e+43) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om))
	tmp = 0
	if l <= -3.2e-32:
		tmp = t_1
	elif l <= -1e-270:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 2.2e+43:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om))
	tmp = 0.0
	if (l <= -3.2e-32)
		tmp = t_1;
	elseif (l <= -1e-270)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 2.2e+43)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	tmp = 0.0;
	if (l <= -3.2e-32)
		tmp = t_1;
	elseif (l <= -1e-270)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 2.2e+43)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.2e-32], t$95$1, If[LessEqual[l, -1e-270], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.2e+43], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2000000000000002e-32 or 2.20000000000000001e43 < l

   1. Initial program 32.2%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in t around 0 43.4%

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

   5. Taylor expanded in U around 0 43.7%

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

   6. Taylor expanded in l around 0 46.3%

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

      1. associate-*r* 46.4%

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

      2. sub-neg 46.4%

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

      3. associate-/l* 45.5%

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

      4. metadata-eval 45.5%

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

   8. Simplified 45.5%

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

      1. associate-*r/ 48.4%

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

      2. *-commutative 48.4%

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

      3. +-commutative 48.4%

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

      4. associate-/r/ 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. associate-*r* 60.4%

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

       2. *-commutative 60.4%

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

       3. +-commutative 60.4%

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

       4. associate-*l/ 60.4%

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

       5. metadata-eval 60.4%

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

       6. sub-neg 60.4%

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

       7. associate-*r* 62.0%

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

       8. *-commutative 62.0%

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

       9. sub-neg 62.0%

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

       10. associate-*l/ 62.0%

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

       11. metadata-eval 62.0%

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

       12. +-commutative 62.0%

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

       13. *-commutative 62.0%

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

   12. Simplified 62.0%

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

   if -3.2000000000000002e-32 < l < -1e-270

   1. Initial program 52.3%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around inf 32.7%

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

      1. *-un-lft-identity 32.7%

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

   6. Applied egg-rr 32.7%

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

      1. *-lft-identity 32.7%

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

      2. associate-*r* 41.8%

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

   8. Simplified 41.8%

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

      1. pow1/2 45.3%

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

      2. *-commutative 45.3%

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

   10. Applied egg-rr 45.3%

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

   if -1e-270 < l < 2.20000000000000001e43

   1. Initial program 71.9%

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

   2. Taylor expanded in Om around inf 61.7%

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

      1. unpow2 61.7%

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

      2. associate-*r/ 61.7%

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

   4. Simplified 61.7%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-270}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 14: 52.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-245}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (/ (* (* (* 2.0 n) l) (* l (* U (+ -2.0 (* U* (/ n Om)))))) Om))))
   (if (<= l -4.6e-32)
     t_1
     (if (<= l 1.6e-245)
       (pow (* 2.0 (* U (* n t))) 0.5)
       (if (<= l 1.15e+28)
         (sqrt (* (* 2.0 n) (* U (+ t (/ n (/ (* Om Om) (* (* l l) U*)))))))
         t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	double tmp;
	if (l <= -4.6e-32) {
		tmp = t_1;
	} else if (l <= 1.6e-245) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 1.15e+28) {
		tmp = sqrt(((2.0 * n) * (U * (t + (n / ((Om * Om) / ((l * l) * U_42_)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((((2.0d0 * n) * l) * (l * (u * ((-2.0d0) + (u_42 * (n / om)))))) / om))
    if (l <= (-4.6d-32)) then
        tmp = t_1
    else if (l <= 1.6d-245) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 1.15d+28) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (n / ((om * om) / ((l * l) * u_42)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	double tmp;
	if (l <= -4.6e-32) {
		tmp = t_1;
	} else if (l <= 1.6e-245) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 1.15e+28) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (n / ((Om * Om) / ((l * l) * U_42_)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om))
	tmp = 0
	if l <= -4.6e-32:
		tmp = t_1
	elif l <= 1.6e-245:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 1.15e+28:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (n / ((Om * Om) / ((l * l) * U_42_)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * l) * Float64(l * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om))
	tmp = 0.0
	if (l <= -4.6e-32)
		tmp = t_1;
	elseif (l <= 1.6e-245)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 1.15e+28)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(n / Float64(Float64(Om * Om) / Float64(Float64(l * l) * U_42_)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((((2.0 * n) * l) * (l * (U * (-2.0 + (U_42_ * (n / Om)))))) / Om));
	tmp = 0.0;
	if (l <= -4.6e-32)
		tmp = t_1;
	elseif (l <= 1.6e-245)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 1.15e+28)
		tmp = sqrt(((2.0 * n) * (U * (t + (n / ((Om * Om) / ((l * l) * U_42_)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * l), $MachinePrecision] * N[(l * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.6e-32], t$95$1, If[LessEqual[l, 1.6e-245], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.15e+28], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.6000000000000001e-32 or 1.14999999999999992e28 < l

   1. Initial program 33.4%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in t around 0 43.5%

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

   5. Taylor expanded in U around 0 43.8%

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

   6. Taylor expanded in l around 0 46.4%

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

      1. associate-*r* 46.4%

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

      2. sub-neg 46.4%

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

      3. associate-/l* 45.6%

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

      4. metadata-eval 45.6%

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

   8. Simplified 45.6%

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

      1. associate-*r/ 48.5%

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

      2. *-commutative 48.5%

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

      3. +-commutative 48.5%

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

      4. associate-/r/ 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. associate-*r* 60.2%

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

       2. *-commutative 60.2%

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

       3. +-commutative 60.2%

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

       4. associate-*l/ 60.2%

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

       5. metadata-eval 60.2%

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

       6. sub-neg 60.2%

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

       7. associate-*r* 61.8%

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

       8. *-commutative 61.8%

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

       9. sub-neg 61.8%

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

       10. associate-*l/ 61.8%

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

       11. metadata-eval 61.8%

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

       12. +-commutative 61.8%

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

       13. *-commutative 61.8%

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

   12. Simplified 61.8%

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

   if -4.6000000000000001e-32 < l < 1.59999999999999993e-245

   1. Initial program 59.0%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in t around inf 44.2%

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

      1. *-un-lft-identity 44.2%

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

   6. Applied egg-rr 44.2%

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

      1. *-lft-identity 44.2%

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

      2. associate-*r* 49.8%

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

   8. Simplified 49.8%

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

      1. pow1/2 52.5%

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

      2. *-commutative 52.5%

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

   10. Applied egg-rr 52.5%

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

   if 1.59999999999999993e-245 < l < 1.14999999999999992e28

   1. Initial program 68.5%

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

      1. associate-*l* 65.4%

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

      2. sub-neg 65.4%

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

      3. associate-+l- 65.4%

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

      4. sub-neg 65.4%

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

      5. associate-/l* 65.4%

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

      6. remove-double-neg 65.4%

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

      7. associate-*l* 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in U* around inf 59.0%

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

      1. mul-1-neg 59.0%

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

      2. associate-/l* 59.0%

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

      3. distribute-neg-frac 59.0%

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

      4. unpow2 59.0%

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

      5. *-commutative 59.0%

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

      6. unpow2 59.0%

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

   6. Simplified 59.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-245}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot n\right) \cdot \ell\right) \cdot \left(\ell \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 15: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.14 \cdot 10^{-250} \lor \neg \left(Om \leq 1.85 \cdot 10^{-7}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.14e-250) (not (<= Om 1.85e-7)))
   (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))
   (sqrt (* 2.0 (/ (* (* U U*) (* (* l l) (* n n))) (* Om Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.14e-250) || !(Om <= 1.85e-7)) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = sqrt((2.0 * (((U * U_42_) * ((l * l) * (n * n))) / (Om * Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.14d-250)) .or. (.not. (om <= 1.85d-7))) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    else
        tmp = sqrt((2.0d0 * (((u * u_42) * ((l * l) * (n * n))) / (om * om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.14e-250) || !(Om <= 1.85e-7)) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * (((U * U_42_) * ((l * l) * (n * n))) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.14e-250) or not (Om <= 1.85e-7):
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	else:
		tmp = math.sqrt((2.0 * (((U * U_42_) * ((l * l) * (n * n))) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.14e-250) || !(Om <= 1.85e-7))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(U * U_42_) * Float64(Float64(l * l) * Float64(n * n))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.14e-250) || ~((Om <= 1.85e-7)))
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	else
		tmp = sqrt((2.0 * (((U * U_42_) * ((l * l) * (n * n))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.14e-250], N[Not[LessEqual[Om, 1.85e-7]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.14000000000000007e-250 or 1.85000000000000002e-7 < Om

   1. Initial program 53.4%

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

   2. Taylor expanded in Om around inf 46.6%

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

      1. unpow2 46.6%

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

      2. associate-*r/ 49.2%

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

   4. Simplified 49.2%

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

   if -1.14000000000000007e-250 < Om < 1.85000000000000002e-7

   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 50.9%

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

   4. Taylor expanded in U* around inf 34.5%

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

      1. associate-*r* 34.5%

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

      2. unpow2 34.5%

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

      3. unpow2 34.5%

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

      4. *-commutative 34.5%

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

      5. unpow2 34.4%

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

   6. Simplified 34.4%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.14 \cdot 10^{-250} \lor \neg \left(Om \leq 1.85 \cdot 10^{-7}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \]

Alternative 16: 46.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.1 \cdot 10^{-250} \lor \neg \left(Om \leq 1.12 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.1e-250) (not (<= Om 1.12e-5)))
   (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))
   (sqrt (* (* 2.0 n) (/ n (/ (* Om Om) (* (* l l) (* U U*))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.1e-250) || !(Om <= 1.12e-5)) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = sqrt(((2.0 * n) * (n / ((Om * Om) / ((l * l) * (U * U_42_))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.1d-250)) .or. (.not. (om <= 1.12d-5))) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    else
        tmp = sqrt(((2.0d0 * n) * (n / ((om * om) / ((l * l) * (u * u_42))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.1e-250) || !(Om <= 1.12e-5)) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (n / ((Om * Om) / ((l * l) * (U * U_42_))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.1e-250) or not (Om <= 1.12e-5):
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (n / ((Om * Om) / ((l * l) * (U * U_42_))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.1e-250) || !(Om <= 1.12e-5))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(n / Float64(Float64(Om * Om) / Float64(Float64(l * l) * Float64(U * U_42_))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.1e-250) || ~((Om <= 1.12e-5)))
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	else
		tmp = sqrt(((2.0 * n) * (n / ((Om * Om) / ((l * l) * (U * U_42_))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.1e-250], N[Not[LessEqual[Om, 1.12e-5]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.1e-250 or 1.11999999999999995e-5 < Om

   1. Initial program 53.4%

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

   2. Taylor expanded in Om around inf 46.6%

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

      1. unpow2 46.6%

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

      2. associate-*r/ 49.2%

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

   4. Simplified 49.2%

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

   if -1.1e-250 < Om < 1.11999999999999995e-5

   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 50.9%

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

   4. Taylor expanded in U* around inf 38.5%

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

      1. associate-/l* 37.2%

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

      2. unpow2 37.2%

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

      3. unpow2 37.2%

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

   6. Simplified 37.2%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.1 \cdot 10^{-250} \lor \neg \left(Om \leq 1.12 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}}}\\ \end{array} \]

Alternative 17: 44.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -5 \cdot 10^{+51} \lor \neg \left(U* \leq 3.3 \cdot 10^{+70}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)}{Om}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -5e+51) (not (<= U* 3.3e+70)))
   (sqrt (* (* 2.0 n) (/ (* l (/ (* n (* l (* U U*))) Om)) Om)))
   (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -5e+51) || !(U_42_ <= 3.3e+70)) {
		tmp = sqrt(((2.0 * n) * ((l * ((n * (l * (U * U_42_))) / Om)) / Om)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-5d+51)) .or. (.not. (u_42 <= 3.3d+70))) then
        tmp = sqrt(((2.0d0 * n) * ((l * ((n * (l * (u * u_42))) / om)) / om)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -5e+51) || !(U_42_ <= 3.3e+70)) {
		tmp = Math.sqrt(((2.0 * n) * ((l * ((n * (l * (U * U_42_))) / Om)) / Om)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -5e+51) or not (U_42_ <= 3.3e+70):
		tmp = math.sqrt(((2.0 * n) * ((l * ((n * (l * (U * U_42_))) / Om)) / Om)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -5e+51) || !(U_42_ <= 3.3e+70))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(l * Float64(Float64(n * Float64(l * Float64(U * U_42_))) / Om)) / Om)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -5e+51) || ~((U_42_ <= 3.3e+70)))
		tmp = sqrt(((2.0 * n) * ((l * ((n * (l * (U * U_42_))) / Om)) / Om)));
	else
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -5e+51], N[Not[LessEqual[U$42$, 3.3e+70]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l * N[(N[(n * N[(l * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -5e51 or 3.30000000000000016e70 < U*

   1. Initial program 45.7%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in t around 0 42.0%

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

   5. Taylor expanded in U around 0 43.8%

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

   6. Taylor expanded in n around -inf 41.5%

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

   if -5e51 < U* < 3.30000000000000016e70

   1. Initial program 53.3%

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

   2. Taylor expanded in Om around inf 50.8%

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

      1. unpow2 50.8%

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

      2. associate-*r/ 52.3%

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

   4. Simplified 52.3%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -5 \cdot 10^{+51} \lor \neg \left(U* \leq 3.3 \cdot 10^{+70}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot U*\right)\right)}{Om}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 18: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-271}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* 2.0 n) (/ (* l (* -2.0 (* U l))) Om)))))
   (if (<= l -1.95e+17)
     t_1
     (if (<= l -8.4e-271)
       (pow (* 2.0 (* U (* n t))) 0.5)
       (if (<= l 2.3e+70) (sqrt (* (* (* 2.0 n) U) t)) t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * ((l * (-2.0 * (U * l))) / Om)));
	double tmp;
	if (l <= -1.95e+17) {
		tmp = t_1;
	} else if (l <= -8.4e-271) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2.3e+70) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * ((l * ((-2.0d0) * (u * l))) / om)))
    if (l <= (-1.95d+17)) then
        tmp = t_1
    else if (l <= (-8.4d-271)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 2.3d+70) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * n) * ((l * (-2.0 * (U * l))) / Om)));
	double tmp;
	if (l <= -1.95e+17) {
		tmp = t_1;
	} else if (l <= -8.4e-271) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2.3e+70) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * ((l * (-2.0 * (U * l))) / Om)))
	tmp = 0
	if l <= -1.95e+17:
		tmp = t_1
	elif l <= -8.4e-271:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 2.3e+70:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(l * Float64(-2.0 * Float64(U * l))) / Om)))
	tmp = 0.0
	if (l <= -1.95e+17)
		tmp = t_1;
	elseif (l <= -8.4e-271)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 2.3e+70)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * ((l * (-2.0 * (U * l))) / Om)));
	tmp = 0.0;
	if (l <= -1.95e+17)
		tmp = t_1;
	elseif (l <= -8.4e-271)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 2.3e+70)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l * N[(-2.0 * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.95e+17], t$95$1, If[LessEqual[l, -8.4e-271], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.3e+70], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.95e17 or 2.29999999999999994e70 < l

   1. Initial program 29.8%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in t around 0 44.2%

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

   5. Taylor expanded in U around 0 44.5%

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

   6. Taylor expanded in n around 0 25.2%

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

   if -1.95e17 < l < -8.4000000000000003e-271

   1. Initial program 51.6%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 29.6%

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

      1. *-un-lft-identity 29.6%

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

   6. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

      2. associate-*r* 39.0%

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

   8. Simplified 39.0%

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

      1. pow1/2 42.1%

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

      2. *-commutative 42.1%

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

   10. Applied egg-rr 42.1%

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

   if -8.4000000000000003e-271 < l < 2.29999999999999994e70

   1. Initial program 71.3%

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

   2. Taylor expanded in t around inf 56.1%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-271}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell \cdot \left(-2 \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \end{array} \]

Alternative 19: 40.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{n \cdot -4}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (/ (* n -4.0) (/ Om (* U (* l l)))))))
   (if (<= l -4.4e+18)
     t_1
     (if (<= l -8.5e-275)
       (pow (* 2.0 (* U (* n t))) 0.5)
       (if (<= l 7.6e+68) (sqrt (* (* (* 2.0 n) U) t)) t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * -4.0) / (Om / (U * (l * l)))));
	double tmp;
	if (l <= -4.4e+18) {
		tmp = t_1;
	} else if (l <= -8.5e-275) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 7.6e+68) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * (-4.0d0)) / (om / (u * (l * l)))))
    if (l <= (-4.4d+18)) then
        tmp = t_1
    else if (l <= (-8.5d-275)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 7.6d+68) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * -4.0) / (Om / (U * (l * l)))));
	double tmp;
	if (l <= -4.4e+18) {
		tmp = t_1;
	} else if (l <= -8.5e-275) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 7.6e+68) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * -4.0) / (Om / (U * (l * l)))))
	tmp = 0
	if l <= -4.4e+18:
		tmp = t_1
	elif l <= -8.5e-275:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 7.6e+68:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * -4.0) / Float64(Om / Float64(U * Float64(l * l)))))
	tmp = 0.0
	if (l <= -4.4e+18)
		tmp = t_1;
	elseif (l <= -8.5e-275)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 7.6e+68)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * -4.0) / (Om / (U * (l * l)))));
	tmp = 0.0;
	if (l <= -4.4e+18)
		tmp = t_1;
	elseif (l <= -8.5e-275)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 7.6e+68)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * -4.0), $MachinePrecision] / N[(Om / N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.4e+18], t$95$1, If[LessEqual[l, -8.5e-275], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 7.6e+68], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.4e18 or 7.6000000000000002e68 < l

   1. Initial program 29.8%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in t around 0 44.2%

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

   5. Taylor expanded in U around 0 44.5%

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

   6. Taylor expanded in n around 0 22.6%

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

      1. associate-/l* 20.6%

         \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

      2. associate-*r/ 20.6%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot n}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

      3. *-commutative 20.6%

         \[\leadsto \sqrt{\frac{-4 \cdot n}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

      4. unpow2 20.6%

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

   8. Simplified 20.6%

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

   if -4.4e18 < l < -8.49999999999999952e-275

   1. Initial program 51.6%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 29.6%

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

      1. *-un-lft-identity 29.6%

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

   6. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

      2. associate-*r* 39.0%

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

   8. Simplified 39.0%

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

      1. pow1/2 42.1%

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

      2. *-commutative 42.1%

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

   10. Applied egg-rr 42.1%

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

   if -8.49999999999999952e-275 < l < 7.6000000000000002e68

   1. Initial program 71.3%

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

   2. Taylor expanded in t around inf 56.1%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\frac{n \cdot -4}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-275}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{n \cdot -4}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \end{array} \]

Alternative 20: 39.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-282}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{n \cdot -4}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -6.8e+15)
   (sqrt (* (* 2.0 n) (* -2.0 (/ (* l l) (/ Om U)))))
   (if (<= l -9.2e-282)
     (pow (* 2.0 (* U (* n t))) 0.5)
     (if (<= l 2e+69)
       (sqrt (* (* (* 2.0 n) U) t))
       (sqrt (/ (* n -4.0) (/ Om (* U (* l l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -6.8e+15) {
		tmp = sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))));
	} else if (l <= -9.2e-282) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2e+69) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt(((n * -4.0) / (Om / (U * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-6.8d+15)) then
        tmp = sqrt(((2.0d0 * n) * ((-2.0d0) * ((l * l) / (om / u)))))
    else if (l <= (-9.2d-282)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (l <= 2d+69) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt(((n * (-4.0d0)) / (om / (u * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -6.8e+15) {
		tmp = Math.sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))));
	} else if (l <= -9.2e-282) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (l <= 2e+69) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt(((n * -4.0) / (Om / (U * (l * l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -6.8e+15:
		tmp = math.sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))))
	elif l <= -9.2e-282:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif l <= 2e+69:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt(((n * -4.0) / (Om / (U * (l * l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -6.8e+15)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(-2.0 * Float64(Float64(l * l) / Float64(Om / U)))));
	elseif (l <= -9.2e-282)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (l <= 2e+69)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(n * -4.0) / Float64(Om / Float64(U * Float64(l * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -6.8e+15)
		tmp = sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))));
	elseif (l <= -9.2e-282)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (l <= 2e+69)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt(((n * -4.0) / (Om / (U * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -6.8e+15], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(-2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -9.2e-282], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2e+69], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * -4.0), $MachinePrecision] / N[(Om / N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.8e15

   1. Initial program 35.7%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in t around 0 42.2%

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

   5. Taylor expanded in n around 0 19.8%

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

      1. *-commutative 19.8%

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

      2. associate-/l* 24.2%

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

      3. unpow2 24.2%

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

   7. Simplified 24.2%

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

   if -6.8e15 < l < -9.1999999999999996e-282

   1. Initial program 51.6%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around inf 29.6%

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

      1. *-un-lft-identity 29.6%

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

   6. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

      2. associate-*r* 39.0%

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

   8. Simplified 39.0%

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

      1. pow1/2 42.1%

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

      2. *-commutative 42.1%

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

   10. Applied egg-rr 42.1%

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

   if -9.1999999999999996e-282 < l < 2.0000000000000001e69

   1. Initial program 71.3%

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

   2. Taylor expanded in t around inf 56.1%

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

   if 2.0000000000000001e69 < l

   1. Initial program 22.0%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in t around 0 46.9%

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

   5. Taylor expanded in U around 0 47.2%

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

   6. Taylor expanded in n around 0 23.9%

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]
   7. Step-by-step derivation

      1. associate-/l* 21.7%

         \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

      2. associate-*r/ 21.7%

         \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot n}{\frac{Om}{{\ell}^{2} \cdot U}}}} \]

      3. *-commutative 21.7%

         \[\leadsto \sqrt{\frac{-4 \cdot n}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

      4. unpow2 21.7%

         \[\leadsto \sqrt{\frac{-4 \cdot n}{\frac{Om}{U \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}} \]

   8. Simplified 21.7%

      \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-282}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{n \cdot -4}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}}\\ \end{array} \]

Alternative 21: 47.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2.5e-12)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.5e-12) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 2.5d-12) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.5e-12) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 2.5e-12:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2.5e-12)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 2.5e-12)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2.5e-12], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.49999999999999985e-12

   1. Initial program 49.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 48.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate-+l- 48.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

      4. sub-neg 48.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      5. associate-/l* 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

      6. remove-double-neg 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

      7. associate-*l* 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 37.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 37.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      2. associate-*r/ 40.4%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   6. Simplified 40.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   if 2.49999999999999985e-12 < t

   1. Initial program 49.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 37.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 37.9%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

   6. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
   7. Step-by-step derivation

      1. *-lft-identity 37.9%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

      2. associate-*r* 44.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]
   9. Step-by-step derivation

      1. pow1/2 47.9%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot t\right) \cdot U\right)\right)}^{0.5}} \]

      2. *-commutative 47.9%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.5} \]

   10. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 22: 47.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;U* \leq -2.05 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= U* -2.05e-212)
     (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 t_1))))
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (U_42_ <= -2.05e-212) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / om)
    if (u_42 <= (-2.05d-212)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * t_1))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (U_42_ <= -2.05e-212) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if U_42_ <= -2.05e-212:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (U_42_ <= -2.05e-212)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * t_1))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (U_42_ <= -2.05e-212)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$42$, -2.05e-212], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.05000000000000007e-212

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf 38.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 38.6%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

      2. associate-*r/ 41.0%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

   4. Simplified 41.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}} \]

   if -2.05000000000000007e-212 < U*

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate-+l- 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

      4. sub-neg 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      5. associate-/l* 54.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

      6. remove-double-neg 54.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

      7. associate-*l* 54.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Simplified 54.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 42.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 42.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      2. associate-*r/ 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   6. Simplified 44.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.05 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 23: 36.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -1.2e-284)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* (* (* 2.0 n) U) t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1.2e-284) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-1.2d-284)) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1.2e-284) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -1.2e-284:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -1.2e-284)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -1.2e-284)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((((2.0 * n) * U) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -1.2e-284], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1.20000000000000001e-284

   1. Initial program 48.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 29.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 29.6%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

   6. Applied egg-rr 29.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
   7. Step-by-step derivation

      1. *-lft-identity 29.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

      2. associate-*r* 34.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

   8. Simplified 34.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]

   if -1.20000000000000001e-284 < U

   1. Initial program 50.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf 31.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \]

Alternative 24: 38.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* U (* n t))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (U * (n * t))), 0.5);
}

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 = (2.0d0 * (u * (n * t))) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((2.0 * (U * (n * t))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (U * (n * t))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * (U * (n * t))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 49.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 29.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
5. Step-by-step derivation

   1. *-un-lft-identity 29.7%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

6. Applied egg-rr 29.7%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
7. Step-by-step derivation

   1. *-lft-identity 29.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

   2. associate-*r* 30.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

8. Simplified 30.8%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]
9. Step-by-step derivation

   1. pow1/2 33.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot t\right) \cdot U\right)\right)}^{0.5}} \]

   2. *-commutative 33.2%

      \[\leadsto {\left(2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.5} \]

10. Applied egg-rr 33.2%

    \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

11. Final simplification 33.2%

    \[\leadsto {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \]

Alternative 25: 36.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* n (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (n * (U * t))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (n * (u * t))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (n * (U * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (n * (U * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(n * Float64(U * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (n * (U * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 29.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

5. Final simplification 29.7%

   \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)} \]

Alternative 26: 36.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 29.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
5. Step-by-step derivation

   1. *-un-lft-identity 29.7%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

6. Applied egg-rr 29.7%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
7. Step-by-step derivation

   1. *-lft-identity 29.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]

   2. associate-*r* 30.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

8. Simplified 30.8%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]

9. Final simplification 30.8%

   \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Reproduce

herbie shell --seed 2023195 
(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*))))))