Toniolo and Linder, Equation (13)

Percentage Accurate: 49.4% → 63.4%

Time: 24.5s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-311}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), \frac{\ell}{Om}, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -5.6e-132)
   (sqrt
    (*
     (* n 2.0)
     (* (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)) U)))
   (if (<= n -4e-311)
     (pow
      (*
       2.0
       (+
        (* n (* t U))
        (/ (+ (* l -2.0) (/ (* (* n l) (- U* U)) Om)) (/ Om (* n (* l U))))))
      0.5)
     (*
      (sqrt (* n 2.0))
      (sqrt
       (* U (fma (fma l -2.0 (* (- U* U) (* n (/ l Om)))) (/ l Om) t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -5.6e-132) {
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (n <= -4e-311) {
		tmp = pow((2.0 * ((n * (t * U)) + (((l * -2.0) + (((n * l) * (U_42_ - U)) / Om)) / (Om / (n * (l * U)))))), 0.5);
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * fma(fma(l, -2.0, ((U_42_ - U) * (n * (l / Om)))), (l / Om), t)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -5.6e-132)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)) * U)));
	elseif (n <= -4e-311)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(Float64(l * -2.0) + Float64(Float64(Float64(n * l) * Float64(U_42_ - U)) / Om)) / Float64(Om / Float64(n * Float64(l * U)))))) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * fma(fma(l, -2.0, Float64(Float64(U_42_ - U) * Float64(n * Float64(l / Om)))), Float64(l / Om), t))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -5.6e-132], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(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] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -4e-311], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(n * l), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(N[(l * -2.0 + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.60000000000000005e-132

   1. Initial program 55.4%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in U around 0 64.3%

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

   if -5.60000000000000005e-132 < n < -3.99999999999979e-311

   1. Initial program 53.0%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in t around inf 59.5%

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

      1. pow1/2 59.6%

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

      2. distribute-lft-out 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-/l* 60.3%

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

      5. associate-*r* 65.7%

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

      6. *-commutative 65.7%

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

      7. *-commutative 65.7%

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

   6. Applied egg-rr 65.7%

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

   if -3.99999999999979e-311 < n

   1. Initial program 47.8%

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

   3. Simplified 60.6%

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

      1. sqrt-prod 70.0%

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

   5. Applied egg-rr 70.0%

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

      1. +-commutative 70.0%

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

      2. *-commutative 70.0%

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

      3. fma-udef 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-*l/ 66.0%

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

      6. associate-*r* 69.1%

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

      7. *-commutative 69.1%

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

      8. associate-*r* 65.2%

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

      9. +-commutative 65.2%

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

      10. fma-def 65.2%

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

   7. Simplified 78.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-311}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), \frac{\ell}{Om}, t\right)}\\ \end{array} \]

Alternative 2: 56.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 - \left(U* \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* n 2.0) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (- U* U) (* n (pow (/ l Om) 2.0)))))))
   (if (<= t_1 2e-319)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (- (* l -2.0) (* (* U* (* n l)) (/ -1.0 Om)))) Om)))))
     (if (<= t_1 5e+300)
       (sqrt t_1)
       (*
        (* l (sqrt 2.0))
        (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 = ((n * 2.0) * U) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * pow((l / Om), 2.0))));
	double tmp;
	if (t_1 <= 2e-319) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	} else if (t_1 <= 5e+300) {
		tmp = sqrt(t_1);
	} else {
		tmp = (l * sqrt(2.0)) * 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) :: tmp
    t_1 = ((n * 2.0d0) * u) * ((t - (2.0d0 * ((l * l) / om))) + ((u_42 - u) * (n * ((l / om) ** 2.0d0))))
    if (t_1 <= 2d-319) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * ((l * (-2.0d0)) - ((u_42 * (n * l)) * ((-1.0d0) / om)))) / om)))))
    else if (t_1 <= 5d+300) then
        tmp = sqrt(t_1)
    else
        tmp = (l * sqrt(2.0d0)) * 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 = ((n * 2.0) * U) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * Math.pow((l / Om), 2.0))));
	double tmp;
	if (t_1 <= 2e-319) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	} else if (t_1 <= 5e+300) {
		tmp = Math.sqrt(t_1);
	} else {
		tmp = (l * Math.sqrt(2.0)) * 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 = ((n * 2.0) * U) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * math.pow((l / Om), 2.0))))
	tmp = 0
	if t_1 <= 2e-319:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))))
	elif t_1 <= 5e+300:
		tmp = math.sqrt(t_1)
	else:
		tmp = (l * math.sqrt(2.0)) * 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(Float64(Float64(n * 2.0) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(U_42_ - U) * Float64(n * (Float64(l / Om) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= 2e-319)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) - Float64(Float64(U_42_ * Float64(n * l)) * Float64(-1.0 / Om)))) / Om)))));
	elseif (t_1 <= 5e+300)
		tmp = sqrt(t_1);
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * 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 = ((n * 2.0) * U) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * ((l / Om) ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= 2e-319)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	elseif (t_1 <= 5e+300)
		tmp = sqrt(t_1);
	else
		tmp = (l * sqrt(2.0)) * 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[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-319], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+300], N[Sqrt[t$95$1], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 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 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*)))) < 1.99998e-319

   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 39.6%

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

   4. Taylor expanded in U around 0 39.5%

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

      1. div-inv 39.5%

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

      2. associate-*r* 41.5%

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

   6. Applied egg-rr 41.5%

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

   if 1.99998e-319 < (*.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*)))) < 5.00000000000000026e300

   1. Initial program 97.9%

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

   if 5.00000000000000026e300 < (*.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 19.4%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in t around inf 46.6%

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

   5. Taylor expanded in l around inf 27.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 2 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 - \left(U* \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 3: 58.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.25e-57)
   (sqrt
    (*
     (* n 2.0)
     (* (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)) U)))
   (if (<= l 4.6e+145)
     (sqrt
      (* (* n 2.0) (* U (+ t (/ (* (* l l) (+ -2.0 (/ n (/ Om U*)))) Om)))))
     (if (<= l 2.5e+198)
       (sqrt
        (+
         (* 2.0 (* n (* t U)))
         (*
          2.0
          (/ (* (+ (* l -2.0) (/ n (/ Om (* l U*)))) (* n (* l U))) Om))))
       (*
        (sqrt 2.0)
        (* l (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 tmp;
	if (l <= 1.25e-57) {
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (l <= 4.6e+145) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else if (l <= 2.5e+198) {
		tmp = sqrt(((2.0 * (n * (t * U))) + (2.0 * ((((l * -2.0) + (n / (Om / (l * U_42_)))) * (n * (l * U))) / Om))));
	} else {
		tmp = sqrt(2.0) * (l * 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) :: tmp
    if (l <= 1.25d-57) then
        tmp = sqrt(((n * 2.0d0) * ((t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)) * u)))
    else if (l <= 4.6d+145) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + (((l * l) * ((-2.0d0) + (n / (om / u_42)))) / om)))))
    else if (l <= 2.5d+198) then
        tmp = sqrt(((2.0d0 * (n * (t * u))) + (2.0d0 * ((((l * (-2.0d0)) + (n / (om / (l * u_42)))) * (n * (l * u))) / om))))
    else
        tmp = sqrt(2.0d0) * (l * 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 tmp;
	if (l <= 1.25e-57) {
		tmp = Math.sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (l <= 4.6e+145) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else if (l <= 2.5e+198) {
		tmp = Math.sqrt(((2.0 * (n * (t * U))) + (2.0 * ((((l * -2.0) + (n / (Om / (l * U_42_)))) * (n * (l * U))) / Om))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.25e-57:
		tmp = math.sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)))
	elif l <= 4.6e+145:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))))
	elif l <= 2.5e+198:
		tmp = math.sqrt(((2.0 * (n * (t * U))) + (2.0 * ((((l * -2.0) + (n / (Om / (l * U_42_)))) * (n * (l * U))) / Om))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.25e-57)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)) * U)));
	elseif (l <= 4.6e+145)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l * l) * Float64(-2.0 + Float64(n / Float64(Om / U_42_)))) / Om)))));
	elseif (l <= 2.5e+198)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(t * U))) + Float64(2.0 * Float64(Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Om / Float64(l * U_42_)))) * Float64(n * Float64(l * U))) / Om))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * 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_)
	tmp = 0.0;
	if (l <= 1.25e-57)
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	elseif (l <= 4.6e+145)
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	elseif (l <= 2.5e+198)
		tmp = sqrt(((2.0 * (n * (t * U))) + (2.0 * ((((l * -2.0) + (n / (Om / (l * U_42_)))) * (n * (l * U))) / Om))));
	else
		tmp = sqrt(2.0) * (l * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.25e-57], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(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] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.6e+145], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.5e+198], N[Sqrt[N[(N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * 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]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.25e-57

   1. Initial program 54.5%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in U around 0 61.2%

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

   if 1.25e-57 < l < 4.6e145

   1. Initial program 52.1%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in U around 0 48.1%

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

   5. Taylor expanded in l around 0 52.7%

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

      1. unpow2 52.7%

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

      2. sub-neg 52.7%

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

      3. associate-/l* 54.8%

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

      4. metadata-eval 54.8%

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

   7. Simplified 54.8%

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

   if 4.6e145 < l < 2.50000000000000024e198

   1. Initial program 33.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in t around inf 84.0%

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

   5. Taylor expanded in U* around inf 84.0%

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

      1. associate-/l* 84.0%

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

   7. Simplified 84.0%

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

   if 2.50000000000000024e198 < l

   1. Initial program 14.7%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in t around 0 51.7%

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

   5. Taylor expanded in l around 0 75.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \left(n \cdot \left(\ell \cdot U\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\right)\\ \end{array} \]

Alternative 4: 57.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\ell \cdot -2 - \frac{n}{\frac{Om}{\ell \cdot U}}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e-59)
   (sqrt
    (*
     (* n 2.0)
     (* (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)) U)))
   (if (<= l 1.15e+140)
     (sqrt
      (* (* n 2.0) (* U (+ t (/ (* (* l l) (+ -2.0 (/ n (/ Om U*)))) Om)))))
     (pow
      (*
       2.0
       (+
        (* n (* t U))
        (/ (- (* l -2.0) (/ n (/ Om (* l U)))) (/ Om (* n (* l U))))))
      0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-59) {
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (l <= 1.15e+140) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else {
		tmp = pow((2.0 * ((n * (t * U)) + (((l * -2.0) - (n / (Om / (l * U)))) / (Om / (n * (l * U)))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2d-59) then
        tmp = sqrt(((n * 2.0d0) * ((t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)) * u)))
    else if (l <= 1.15d+140) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + (((l * l) * ((-2.0d0) + (n / (om / u_42)))) / om)))))
    else
        tmp = (2.0d0 * ((n * (t * u)) + (((l * (-2.0d0)) - (n / (om / (l * u)))) / (om / (n * (l * u)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-59) {
		tmp = Math.sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (l <= 1.15e+140) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else {
		tmp = Math.pow((2.0 * ((n * (t * U)) + (((l * -2.0) - (n / (Om / (l * U)))) / (Om / (n * (l * U)))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2e-59:
		tmp = math.sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)))
	elif l <= 1.15e+140:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))))
	else:
		tmp = math.pow((2.0 * ((n * (t * U)) + (((l * -2.0) - (n / (Om / (l * U)))) / (Om / (n * (l * U)))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2e-59)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)) * U)));
	elseif (l <= 1.15e+140)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l * l) * Float64(-2.0 + Float64(n / Float64(Om / U_42_)))) / Om)))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(Float64(l * -2.0) - Float64(n / Float64(Om / Float64(l * U)))) / Float64(Om / Float64(n * Float64(l * U)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2e-59)
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	elseif (l <= 1.15e+140)
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	else
		tmp = (2.0 * ((n * (t * U)) + (((l * -2.0) - (n / (Om / (l * U)))) / (Om / (n * (l * U)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2e-59], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(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] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.15e+140], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] - N[(n / N[(Om / N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.0000000000000001e-59

   1. Initial program 54.5%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in U around 0 61.2%

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

   if 2.0000000000000001e-59 < l < 1.14999999999999995e140

   1. Initial program 53.3%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in U around 0 49.1%

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

   5. Taylor expanded in l around 0 53.8%

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

      1. unpow2 53.8%

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

      2. sub-neg 53.8%

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

      3. associate-/l* 56.0%

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

      4. metadata-eval 56.0%

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

   7. Simplified 56.0%

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

   if 1.14999999999999995e140 < l

   1. Initial program 19.1%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in t around inf 58.2%

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

      1. pow1/2 58.5%

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

      2. distribute-lft-out 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-/l* 54.0%

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

      5. associate-*r* 54.2%

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

      6. *-commutative 54.2%

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

      7. *-commutative 54.2%

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

   6. Applied egg-rr 54.2%

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

   7. Taylor expanded in U* around 0 58.0%

      \[\leadsto {\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\color{blue}{-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \]
   8. Step-by-step derivation

      1. mul-1-neg 58.0%

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

      2. associate-/l* 58.0%

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

      3. distribute-neg-frac 58.0%

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

   9. Simplified 58.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\ell \cdot -2 - \frac{n}{\frac{Om}{\ell \cdot U}}}{\frac{Om}{n \cdot \left(\ell \cdot U\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 5: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n}{\frac{Om}{U*}}\\ \mathbf{if}\;\ell \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + t_1\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - t_1\right) \cdot \left(\frac{n \cdot -2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ n (/ Om U*))))
   (if (<= l 2e-12)
     (sqrt (* (* n 2.0) (* U (+ t (/ (/ n (/ Om (* U* (* l l)))) Om)))))
     (if (<= l 5e+145)
       (sqrt (* (* n 2.0) (* U (+ t (/ (* (* l l) (+ -2.0 t_1)) Om)))))
       (sqrt (* (- 2.0 t_1) (* (/ (* n -2.0) Om) (* l (* l U)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n / (Om / U_42_);
	double tmp;
	if (l <= 2e-12) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else if (l <= 5e+145) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + t_1)) / Om)))));
	} else {
		tmp = sqrt(((2.0 - t_1) * (((n * -2.0) / Om) * (l * (l * 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 / (om / u_42)
    if (l <= 2d-12) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((n / (om / (u_42 * (l * l)))) / om)))))
    else if (l <= 5d+145) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + (((l * l) * ((-2.0d0) + t_1)) / om)))))
    else
        tmp = sqrt(((2.0d0 - t_1) * (((n * (-2.0d0)) / om) * (l * (l * 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 / (Om / U_42_);
	double tmp;
	if (l <= 2e-12) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else if (l <= 5e+145) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + t_1)) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 - t_1) * (((n * -2.0) / Om) * (l * (l * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n / (Om / U_42_)
	tmp = 0
	if l <= 2e-12:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))))
	elif l <= 5e+145:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + t_1)) / Om)))))
	else:
		tmp = math.sqrt(((2.0 - t_1) * (((n * -2.0) / Om) * (l * (l * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n / Float64(Om / U_42_))
	tmp = 0.0
	if (l <= 2e-12)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n / Float64(Om / Float64(U_42_ * Float64(l * l)))) / Om)))));
	elseif (l <= 5e+145)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l * l) * Float64(-2.0 + t_1)) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 - t_1) * Float64(Float64(Float64(n * -2.0) / Om) * Float64(l * Float64(l * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n / (Om / U_42_);
	tmp = 0.0;
	if (l <= 2e-12)
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	elseif (l <= 5e+145)
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + t_1)) / Om)))));
	else
		tmp = sqrt(((2.0 - t_1) * (((n * -2.0) / Om) * (l * (l * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2e-12], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n / N[(Om / N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5e+145], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] * N[(-2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 - t$95$1), $MachinePrecision] * N[(N[(N[(n * -2.0), $MachinePrecision] / Om), $MachinePrecision] * N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.99999999999999996e-12

   1. Initial program 54.4%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in U around 0 59.8%

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

   5. Taylor expanded in n around inf 53.9%

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

      1. associate-/l* 54.4%

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

      2. *-commutative 54.4%

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

      3. unpow2 54.4%

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

   7. Simplified 54.4%

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

   if 1.99999999999999996e-12 < l < 4.99999999999999967e145

   1. Initial program 51.8%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in U around 0 52.3%

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

   5. Taylor expanded in l around 0 58.5%

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

      1. unpow2 58.5%

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

      2. sub-neg 58.5%

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

      3. associate-/l* 61.5%

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

      4. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

   if 4.99999999999999967e145 < l

   1. Initial program 19.9%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in l around -inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. associate-*r/ 43.6%

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

      3. *-commutative 43.6%

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

      4. associate-/r* 43.3%

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

      5. *-commutative 43.3%

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

      6. associate-/r* 42.9%

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

      7. mul-1-neg 42.9%

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

      8. unsub-neg 42.9%

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

      9. associate-/l* 42.9%

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

      10. unpow2 42.9%

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

   6. Simplified 42.9%

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

      1. *-un-lft-identity 42.9%

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

      2. associate-/l/ 42.9%

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

      3. associate-/r/ 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

      2. unpow2 42.9%

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

      3. associate-/r* 42.9%

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

      4. unpow2 42.9%

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

   10. Simplified 42.9%

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

       1. *-un-lft-identity 42.9%

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

       2. associate-/r/ 42.9%

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

       3. *-commutative 42.9%

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

       4. associate-/l/ 43.6%

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

       5. associate-*l/ 43.6%

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

   12. Applied egg-rr 43.6%

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

       1. *-lft-identity 43.6%

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

       2. *-commutative 43.6%

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

       3. associate-/l* 43.6%

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

       4. associate-/r/ 43.0%

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

       5. associate-*l* 56.2%

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

   14. Simplified 56.2%

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

   15. Taylor expanded in U* around inf 47.4%

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

       1. associate-/l* 56.2%

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

   17. Simplified 56.2%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U*}}\right) \cdot \left(\frac{n \cdot -2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \]

Alternative 6: 53.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U* - U}}\right) \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(-2 \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5e-10)
   (sqrt (* (* n 2.0) (* U (+ t (/ (/ n (/ Om (* U* (* l l)))) Om)))))
   (if (<= l 5e+145)
     (sqrt
      (* (* n 2.0) (* U (+ t (/ (* (* l l) (+ -2.0 (/ n (/ Om U*)))) Om)))))
     (sqrt
      (* (- 2.0 (/ n (/ Om (- U* U)))) (* l (* (* l U) (* -2.0 (/ n Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5e-10) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else if (l <= 5e+145) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else {
		tmp = sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / 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 <= 5d-10) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((n / (om / (u_42 * (l * l)))) / om)))))
    else if (l <= 5d+145) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + (((l * l) * ((-2.0d0) + (n / (om / u_42)))) / om)))))
    else
        tmp = sqrt(((2.0d0 - (n / (om / (u_42 - u)))) * (l * ((l * u) * ((-2.0d0) * (n / 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 <= 5e-10) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else if (l <= 5e+145) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5e-10:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))))
	elif l <= 5e+145:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5e-10)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n / Float64(Om / Float64(U_42_ * Float64(l * l)))) / Om)))));
	elseif (l <= 5e+145)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l * l) * Float64(-2.0 + Float64(n / Float64(Om / U_42_)))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 - Float64(n / Float64(Om / Float64(U_42_ - U)))) * Float64(l * Float64(Float64(l * U) * Float64(-2.0 * Float64(n / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5e-10)
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	elseif (l <= 5e+145)
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	else
		tmp = sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5e-10], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n / N[(Om / N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5e+145], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 - N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(l * U), $MachinePrecision] * N[(-2.0 * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.00000000000000031e-10

   1. Initial program 54.4%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in U around 0 59.8%

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

   5. Taylor expanded in n around inf 53.9%

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

      1. associate-/l* 54.4%

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

      2. *-commutative 54.4%

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

      3. unpow2 54.4%

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

   7. Simplified 54.4%

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

   if 5.00000000000000031e-10 < l < 4.99999999999999967e145

   1. Initial program 51.8%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in U around 0 52.3%

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

   5. Taylor expanded in l around 0 58.5%

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

      1. unpow2 58.5%

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

      2. sub-neg 58.5%

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

      3. associate-/l* 61.5%

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

      4. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

   if 4.99999999999999967e145 < l

   1. Initial program 19.9%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in l around -inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. associate-*r/ 43.6%

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

      3. *-commutative 43.6%

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

      4. associate-/r* 43.3%

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

      5. *-commutative 43.3%

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

      6. associate-/r* 42.9%

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

      7. mul-1-neg 42.9%

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

      8. unsub-neg 42.9%

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

      9. associate-/l* 42.9%

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

      10. unpow2 42.9%

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

   6. Simplified 42.9%

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

      1. *-un-lft-identity 42.9%

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

      2. associate-/l/ 42.9%

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

      3. associate-/r/ 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

      2. unpow2 42.9%

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

      3. associate-/r* 42.9%

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

      4. unpow2 42.9%

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

   10. Simplified 42.9%

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

       1. *-un-lft-identity 42.9%

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

       2. associate-/r/ 42.9%

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

       3. *-commutative 42.9%

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

       4. associate-/l/ 43.6%

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

       5. associate-*l/ 43.6%

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

   12. Applied egg-rr 43.6%

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

       1. *-lft-identity 43.6%

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

       2. *-commutative 43.6%

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

       3. associate-/l* 43.6%

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

       4. associate-/r/ 43.0%

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

       5. associate-*l* 56.2%

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

   14. Simplified 56.2%

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

       1. pow1 56.2%

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

       2. *-commutative 56.2%

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

       3. associate-/l* 56.2%

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

   16. Applied egg-rr 56.2%

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

       1. unpow1 56.2%

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

       2. associate-*l* 65.3%

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

       3. associate-/r/ 65.3%

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

   18. Simplified 65.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U* - U}}\right) \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(-2 \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 7: 57.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-59}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U* - U}}\right) \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(-2 \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e-59)
   (sqrt
    (*
     (* n 2.0)
     (* (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)) U)))
   (if (<= l 3.6e+145)
     (sqrt
      (* (* n 2.0) (* U (+ t (/ (* (* l l) (+ -2.0 (/ n (/ Om U*)))) Om)))))
     (sqrt
      (* (- 2.0 (/ n (/ Om (- U* U)))) (* l (* (* l U) (* -2.0 (/ n Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e-59) {
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (l <= 3.6e+145) {
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else {
		tmp = sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / 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 <= 1d-59) then
        tmp = sqrt(((n * 2.0d0) * ((t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)) * u)))
    else if (l <= 3.6d+145) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + (((l * l) * ((-2.0d0) + (n / (om / u_42)))) / om)))))
    else
        tmp = sqrt(((2.0d0 - (n / (om / (u_42 - u)))) * (l * ((l * u) * ((-2.0d0) * (n / 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 <= 1e-59) {
		tmp = Math.sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	} else if (l <= 3.6e+145) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1e-59:
		tmp = math.sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)))
	elif l <= 3.6e+145:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e-59)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)) * U)));
	elseif (l <= 3.6e+145)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l * l) * Float64(-2.0 + Float64(n / Float64(Om / U_42_)))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 - Float64(n / Float64(Om / Float64(U_42_ - U)))) * Float64(l * Float64(Float64(l * U) * Float64(-2.0 * Float64(n / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1e-59)
		tmp = sqrt(((n * 2.0) * ((t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)) * U)));
	elseif (l <= 3.6e+145)
		tmp = sqrt(((n * 2.0) * (U * (t + (((l * l) * (-2.0 + (n / (Om / U_42_)))) / Om)))));
	else
		tmp = sqrt(((2.0 - (n / (Om / (U_42_ - U)))) * (l * ((l * U) * (-2.0 * (n / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e-59], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(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] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.6e+145], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 - N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(l * U), $MachinePrecision] * N[(-2.0 * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1e-59

   1. Initial program 54.5%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in U around 0 61.2%

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

   if 1e-59 < l < 3.59999999999999974e145

   1. Initial program 52.1%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in U around 0 48.1%

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

   5. Taylor expanded in l around 0 52.7%

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

      1. unpow2 52.7%

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

      2. sub-neg 52.7%

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

      3. associate-/l* 54.8%

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

      4. metadata-eval 54.8%

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

   7. Simplified 54.8%

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

   if 3.59999999999999974e145 < l

   1. Initial program 19.9%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in l around -inf 43.6%

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

      1. associate-/l* 43.6%

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

      2. associate-*r/ 43.6%

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

      3. *-commutative 43.6%

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

      4. associate-/r* 43.3%

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

      5. *-commutative 43.3%

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

      6. associate-/r* 42.9%

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

      7. mul-1-neg 42.9%

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

      8. unsub-neg 42.9%

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

      9. associate-/l* 42.9%

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

      10. unpow2 42.9%

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

   6. Simplified 42.9%

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

      1. *-un-lft-identity 42.9%

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

      2. associate-/l/ 42.9%

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

      3. associate-/r/ 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

      2. unpow2 42.9%

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

      3. associate-/r* 42.9%

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

      4. unpow2 42.9%

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

   10. Simplified 42.9%

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

       1. *-un-lft-identity 42.9%

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

       2. associate-/r/ 42.9%

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

       3. *-commutative 42.9%

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

       4. associate-/l/ 43.6%

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

       5. associate-*l/ 43.6%

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

   12. Applied egg-rr 43.6%

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

       1. *-lft-identity 43.6%

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

       2. *-commutative 43.6%

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

       3. associate-/l* 43.6%

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

       4. associate-/r/ 43.0%

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

       5. associate-*l* 56.2%

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

   14. Simplified 56.2%

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

       1. pow1 56.2%

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

       2. *-commutative 56.2%

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

       3. associate-/l* 56.2%

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

   16. Applied egg-rr 56.2%

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

       1. unpow1 56.2%

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

       2. associate-*l* 65.3%

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

       3. associate-/r/ 65.3%

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

   18. Simplified 65.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-59}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U* - U}}\right) \cdot \left(\ell \cdot \left(\left(\ell \cdot U\right) \cdot \left(-2 \cdot \frac{n}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 50.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e+95)
   (sqrt (* (* n 2.0) (* U (+ t (/ (/ n (/ Om (* U* (* l l)))) Om)))))
   (pow (* -4.0 (/ n (/ Om (* l (* l U))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+95) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else {
		tmp = pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.15d+95) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((n / (om / (u_42 * (l * l)))) / om)))))
    else
        tmp = ((-4.0d0) * (n / (om / (l * (l * u))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+95) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else {
		tmp = Math.pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e+95:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))))
	else:
		tmp = math.pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e+95)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n / Float64(Om / Float64(U_42_ * Float64(l * l)))) / Om)))));
	else
		tmp = Float64(-4.0 * Float64(n / Float64(Om / Float64(l * Float64(l * U))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e+95)
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	else
		tmp = (-4.0 * (n / (Om / (l * (l * U))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e+95], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n / N[(Om / N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(n / N[(Om / N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.14999999999999999e95

   1. Initial program 54.4%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in U around 0 58.9%

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

   5. Taylor expanded in n around inf 53.3%

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

      1. associate-/l* 54.2%

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

      2. *-commutative 54.2%

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

      3. unpow2 54.2%

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

   7. Simplified 54.2%

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

   if 1.14999999999999999e95 < l

   1. Initial program 29.0%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in l around -inf 47.7%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-/r* 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-/r* 50.2%

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

      7. mul-1-neg 50.2%

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

      8. unsub-neg 50.2%

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

      9. associate-/l* 50.2%

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

      10. unpow2 50.2%

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

   6. Simplified 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-/l/ 50.2%

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

      3. associate-/r/ 50.2%

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

   8. Applied egg-rr 50.2%

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

      1. *-lft-identity 50.2%

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

      2. unpow2 50.2%

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

      3. associate-/r* 50.2%

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

      4. unpow2 50.2%

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

   10. Simplified 50.2%

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

   11. Taylor expanded in Om around inf 17.4%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{{\ell}^{2} \cdot U}}}} \]
   12. Step-by-step derivation

       1. unpow2 17.4%

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

       2. associate-*l* 20.3%

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

   13. Simplified 20.3%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}} \]
   14. Step-by-step derivation

       1. pow1/2 44.9%

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

       2. times-frac 44.9%

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

       3. metadata-eval 44.9%

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

   15. Applied egg-rr 44.9%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 9: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U*}}\right) \cdot \left(\frac{n \cdot -2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e+94)
   (sqrt (* (* n 2.0) (* U (+ t (/ (/ n (/ Om (* U* (* l l)))) Om)))))
   (sqrt (* (- 2.0 (/ n (/ Om U*))) (* (/ (* n -2.0) Om) (* l (* l U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.5e+94) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else {
		tmp = sqrt(((2.0 - (n / (Om / U_42_))) * (((n * -2.0) / Om) * (l * (l * U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.5d+94) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((n / (om / (u_42 * (l * l)))) / om)))))
    else
        tmp = sqrt(((2.0d0 - (n / (om / u_42))) * (((n * (-2.0d0)) / om) * (l * (l * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.5e+94) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 - (n / (Om / U_42_))) * (((n * -2.0) / Om) * (l * (l * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.5e+94:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 - (n / (Om / U_42_))) * (((n * -2.0) / Om) * (l * (l * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.5e+94)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n / Float64(Om / Float64(U_42_ * Float64(l * l)))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 - Float64(n / Float64(Om / U_42_))) * Float64(Float64(Float64(n * -2.0) / Om) * Float64(l * Float64(l * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.5e+94)
		tmp = sqrt(((n * 2.0) * (U * (t + ((n / (Om / (U_42_ * (l * l)))) / Om)))));
	else
		tmp = sqrt(((2.0 - (n / (Om / U_42_))) * (((n * -2.0) / Om) * (l * (l * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.5e+94], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n / N[(Om / N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 - N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * -2.0), $MachinePrecision] / Om), $MachinePrecision] * N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.49999999999999972e94

   1. Initial program 54.4%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in U around 0 58.9%

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

   5. Taylor expanded in n around inf 53.3%

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

      1. associate-/l* 54.2%

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

      2. *-commutative 54.2%

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

      3. unpow2 54.2%

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

   7. Simplified 54.2%

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

   if 4.49999999999999972e94 < l

   1. Initial program 29.0%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in l around -inf 47.7%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-/r* 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-/r* 50.2%

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

      7. mul-1-neg 50.2%

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

      8. unsub-neg 50.2%

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

      9. associate-/l* 50.2%

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

      10. unpow2 50.2%

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

   6. Simplified 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-/l/ 50.2%

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

      3. associate-/r/ 50.2%

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

   8. Applied egg-rr 50.2%

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

      1. *-lft-identity 50.2%

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

      2. unpow2 50.2%

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

      3. associate-/r* 50.2%

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

      4. unpow2 50.2%

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

   10. Simplified 50.2%

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

       1. *-un-lft-identity 50.2%

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

       2. associate-/r/ 50.2%

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

       3. *-commutative 50.2%

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

       4. associate-/l/ 47.7%

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

       5. associate-*l/ 47.7%

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

   12. Applied egg-rr 47.7%

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

       1. *-lft-identity 47.7%

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

       2. *-commutative 47.7%

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

       3. associate-/l* 47.7%

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

       4. associate-/r/ 47.2%

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

       5. associate-*l* 55.9%

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

   14. Simplified 55.9%

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

   15. Taylor expanded in U* around inf 50.1%

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

       1. associate-/l* 55.9%

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

   17. Simplified 55.9%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\frac{n}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - \frac{n}{\frac{Om}{U*}}\right) \cdot \left(\frac{n \cdot -2}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \]

Alternative 10: 48.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5e+104)
   (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (pow (* -4.0 (/ n (/ Om (* l (* l U))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5e+104) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 5d+104) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = ((-4.0d0) * (n / (om / (l * (l * u))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5e+104) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5e+104:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5e+104)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(-4.0 * Float64(n / Float64(Om / Float64(l * Float64(l * U))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5e+104)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = (-4.0 * (n / (Om / (l * (l * U))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5e+104], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(n / N[(Om / N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999997e104

   1. Initial program 54.2%

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

      1. associate-*l* 55.9%

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

      2. sub-neg 55.9%

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

      3. associate-+l- 55.9%

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

      4. sub-neg 55.9%

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

      5. associate-/l* 59.8%

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

      6. remove-double-neg 59.8%

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

      7. associate-*l* 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in Om around inf 47.6%

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

      1. unpow2 47.6%

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

      2. associate-*r/ 51.6%

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

   6. Simplified 51.6%

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

   if 4.9999999999999997e104 < l

   1. Initial program 29.7%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in l around -inf 49.0%

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

      1. associate-/l* 49.0%

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

      2. associate-*r/ 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/r* 51.9%

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

      5. *-commutative 51.9%

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

      6. associate-/r* 51.7%

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

      7. mul-1-neg 51.7%

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

      8. unsub-neg 51.7%

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

      9. associate-/l* 51.7%

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

      10. unpow2 51.7%

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

   6. Simplified 51.7%

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

      1. *-un-lft-identity 51.7%

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

      2. associate-/l/ 51.7%

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

      3. associate-/r/ 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. *-lft-identity 51.7%

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

      2. unpow2 51.7%

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

      3. associate-/r* 51.7%

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

      4. unpow2 51.7%

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

   10. Simplified 51.7%

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

   11. Taylor expanded in Om around inf 17.7%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{{\ell}^{2} \cdot U}}}} \]
   12. Step-by-step derivation

       1. unpow2 17.7%

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

       2. associate-*l* 20.8%

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

   13. Simplified 20.8%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}} \]
   14. Step-by-step derivation

       1. pow1/2 46.1%

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

       2. times-frac 46.1%

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

       3. metadata-eval 46.1%

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

   15. Applied egg-rr 46.1%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 11: 42.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{+86}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.6e+86)
   (pow (* 2.0 (* n (* t U))) 0.5)
   (pow (* -4.0 (/ n (/ Om (* l (* l U))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.6e+86) {
		tmp = pow((2.0 * (n * (t * U))), 0.5);
	} else {
		tmp = pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 8.6d+86) then
        tmp = (2.0d0 * (n * (t * u))) ** 0.5d0
    else
        tmp = ((-4.0d0) * (n / (om / (l * (l * u))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.6e+86) {
		tmp = Math.pow((2.0 * (n * (t * U))), 0.5);
	} else {
		tmp = Math.pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.6e+86:
		tmp = math.pow((2.0 * (n * (t * U))), 0.5)
	else:
		tmp = math.pow((-4.0 * (n / (Om / (l * (l * U))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.6e+86)
		tmp = Float64(2.0 * Float64(n * Float64(t * U))) ^ 0.5;
	else
		tmp = Float64(-4.0 * Float64(n / Float64(Om / Float64(l * Float64(l * U))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8.6e+86)
		tmp = (2.0 * (n * (t * U))) ^ 0.5;
	else
		tmp = (-4.0 * (n / (Om / (l * (l * U))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.6e+86], N[Power[N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(-4.0 * N[(n / N[(Om / N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.6000000000000004e86

   1. Initial program 54.2%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in t around inf 43.7%

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

      1. pow1/2 44.6%

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

      2. associate-*r* 44.6%

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

      3. *-commutative 44.6%

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

   6. Applied egg-rr 44.6%

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

   if 8.6000000000000004e86 < l

   1. Initial program 32.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in l around -inf 46.8%

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

      1. associate-/l* 46.8%

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

      2. associate-*r/ 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-/r* 49.3%

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

      5. *-commutative 49.3%

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

      6. associate-/r* 49.1%

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

      7. mul-1-neg 49.1%

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

      8. unsub-neg 49.1%

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

      9. associate-/l* 49.1%

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

      10. unpow2 49.1%

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

   6. Simplified 49.1%

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

      1. *-un-lft-identity 49.1%

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

      2. associate-/l/ 49.1%

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

      3. associate-/r/ 49.1%

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

   8. Applied egg-rr 49.1%

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

      1. *-lft-identity 49.1%

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

      2. unpow2 49.1%

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

      3. associate-/r* 49.1%

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

      4. unpow2 49.1%

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

   10. Simplified 49.1%

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

   11. Taylor expanded in Om around inf 18.7%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{{\ell}^{2} \cdot U}}}} \]
   12. Step-by-step derivation

       1. unpow2 18.7%

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

       2. associate-*l* 21.4%

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

   13. Simplified 21.4%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}} \]
   14. Step-by-step derivation

       1. pow1/2 44.0%

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

       2. times-frac 44.0%

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

       3. metadata-eval 44.0%

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

   15. Applied egg-rr 44.0%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{+86}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 12: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{+94}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.6e+94)
   (pow (* 2.0 (* n (* t U))) 0.5)
   (sqrt (* -4.0 (* (/ n Om) (* l (* l U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.6e+94) {
		tmp = pow((2.0 * (n * (t * U))), 0.5);
	} else {
		tmp = sqrt((-4.0 * ((n / Om) * (l * (l * U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.6d+94) then
        tmp = (2.0d0 * (n * (t * u))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * ((n / om) * (l * (l * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.6e+94) {
		tmp = Math.pow((2.0 * (n * (t * U))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * ((n / Om) * (l * (l * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.6e+94:
		tmp = math.pow((2.0 * (n * (t * U))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * ((n / Om) * (l * (l * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.6e+94)
		tmp = Float64(2.0 * Float64(n * Float64(t * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(n / Om) * Float64(l * Float64(l * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.6e+94)
		tmp = (2.0 * (n * (t * U))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * ((n / Om) * (l * (l * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.6e+94], N[Power[N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(n / Om), $MachinePrecision] * N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.6e94

   1. Initial program 54.4%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 44.5%

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

      2. associate-*r* 44.5%

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

      3. *-commutative 44.5%

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

   6. Applied egg-rr 44.5%

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

   if 6.6e94 < l

   1. Initial program 29.0%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in l around -inf 47.7%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-/r* 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-/r* 50.2%

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

      7. mul-1-neg 50.2%

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

      8. unsub-neg 50.2%

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

      9. associate-/l* 50.2%

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

      10. unpow2 50.2%

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

   6. Simplified 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-/l/ 50.2%

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

      3. associate-/r/ 50.2%

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

   8. Applied egg-rr 50.2%

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

      1. *-lft-identity 50.2%

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

      2. unpow2 50.2%

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

      3. associate-/r* 50.2%

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

      4. unpow2 50.2%

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

   10. Simplified 50.2%

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

   11. Taylor expanded in Om around inf 17.4%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{{\ell}^{2} \cdot U}}}} \]
   12. Step-by-step derivation

       1. unpow2 17.4%

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

       2. associate-*l* 20.3%

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

   13. Simplified 20.3%

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

   14. Taylor expanded in n around 0 17.4%

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

       1. unpow2 17.4%

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

       2. associate-*r* 20.3%

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

       3. associate-*l/ 19.7%

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

   16. Simplified 19.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{+94}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array} \]

Alternative 13: 40.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4e+95)
   (pow (* 2.0 (* n (* t U))) 0.5)
   (sqrt (* -4.0 (/ n (/ Om (* l (* l U))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e+95) {
		tmp = pow((2.0 * (n * (t * U))), 0.5);
	} else {
		tmp = sqrt((-4.0 * (n / (Om / (l * (l * U))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4d+95) then
        tmp = (2.0d0 * (n * (t * u))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * (n / (om / (l * (l * u))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e+95) {
		tmp = Math.pow((2.0 * (n * (t * U))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * (n / (Om / (l * (l * U))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4e+95:
		tmp = math.pow((2.0 * (n * (t * U))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * (n / (Om / (l * (l * U))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4e+95)
		tmp = Float64(2.0 * Float64(n * Float64(t * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(n / Float64(Om / Float64(l * Float64(l * U))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4e+95)
		tmp = (2.0 * (n * (t * U))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * (n / (Om / (l * (l * U))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4e+95], N[Power[N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(n / N[(Om / N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000008e95

   1. Initial program 54.4%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 44.5%

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

      2. associate-*r* 44.5%

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

      3. *-commutative 44.5%

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

   6. Applied egg-rr 44.5%

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

   if 4.00000000000000008e95 < l

   1. Initial program 29.0%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in l around -inf 47.7%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-/r* 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-/r* 50.2%

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

      7. mul-1-neg 50.2%

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

      8. unsub-neg 50.2%

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

      9. associate-/l* 50.2%

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

      10. unpow2 50.2%

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

   6. Simplified 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-/l/ 50.2%

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

      3. associate-/r/ 50.2%

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

   8. Applied egg-rr 50.2%

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

      1. *-lft-identity 50.2%

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

      2. unpow2 50.2%

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

      3. associate-/r* 50.2%

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

      4. unpow2 50.2%

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

   10. Simplified 50.2%

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

   11. Taylor expanded in Om around inf 17.4%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{{\ell}^{2} \cdot U}}}} \]
   12. Step-by-step derivation

       1. unpow2 17.4%

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

       2. associate-*l* 20.3%

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

   13. Simplified 20.3%

       \[\leadsto \sqrt{\frac{-2 \cdot n}{\color{blue}{0.5 \cdot \frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}} \]
   14. Step-by-step derivation

       1. times-frac 20.3%

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

       2. metadata-eval 20.3%

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

   15. Applied egg-rr 20.3%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\ell \cdot U\right)}}}\\ \end{array} \]

Alternative 14: 37.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* n (* t U))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (n * (t * U))), 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 * (n * (t * u))) ** 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 * (n * (t * U))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (n * (t * U))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(n * Float64(t * U))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * (n * (t * U))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 51.1%

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

3. Simplified 60.9%

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

4. Taylor expanded in t around inf 38.6%

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

   1. pow1/2 39.4%

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

   2. associate-*r* 39.4%

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

   3. *-commutative 39.4%

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

6. Applied egg-rr 39.4%

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

7. Final simplification 39.4%

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

Alternative 15: 35.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

3. Simplified 60.9%

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

4. Taylor expanded in t around inf 38.6%

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

5. Final simplification 38.6%

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

Reproduce

herbie shell --seed 2023258 
(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*))))))