Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 59.4%

Time: 22.8s

Alternatives: 18

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.6% 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: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 + U* \cdot \frac{n}{Om}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)}\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot t_1\right)}}}\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{-191}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot t_1}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ -2.0 (* U* (/ n Om))))
        (t_2
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ l (/ Om (fma l -2.0 (/ n (/ Om (* l U*))))))))))))
   (if (<= l -2.2e+155)
     (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l t_1))))))
     (if (<= l -7.8e-191)
       t_2
       (if (<= l 2.4e-102)
         (sqrt (* 2.0 (* U (* n (+ t (/ (* (* l l) -2.0) Om))))))
         (if (<= l 1.7e+129)
           t_2
           (* (sqrt 2.0) (* l (sqrt (/ n (/ Om (* U t_1))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = -2.0 + (U_42_ * (n / Om));
	double t_2 = sqrt(((2.0 * n) * (U * (t + (l / (Om / fma(l, -2.0, (n / (Om / (l * U_42_))))))))));
	double tmp;
	if (l <= -2.2e+155) {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * t_1))))));
	} else if (l <= -7.8e-191) {
		tmp = t_2;
	} else if (l <= 2.4e-102) {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	} else if (l <= 1.7e+129) {
		tmp = t_2;
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * t_1)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))
	t_2 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(l / Float64(Om / fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_))))))))))
	tmp = 0.0
	if (l <= -2.2e+155)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * t_1))))));
	elseif (l <= -7.8e-191)
		tmp = t_2;
	elseif (l <= 2.4e-102)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) * -2.0) / Om))))));
	elseif (l <= 1.7e+129)
		tmp = t_2;
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * t_1))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(l / N[(Om / N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.2e+155], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -7.8e-191], t$95$2, If[LessEqual[l, 2.4e-102], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.7e+129], t$95$2, N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.2000000000000002e155

   1. Initial program 20.9%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in U around 0 43.7%

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

      1. associate-*r* 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 53.7%

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

      4. *-commutative 53.7%

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

      5. fma-def 53.7%

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

      6. associate-/l* 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in t around 0 50.4%

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

      1. associate-/l* 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*l/ 60.4%

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

      4. *-commutative 60.4%

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

      5. fma-udef 60.4%

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

      6. *-commutative 60.4%

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

   9. Simplified 60.4%

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

   10. Taylor expanded in l around 0 67.3%

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

       1. associate-*r* 67.3%

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

       2. sub-neg 67.3%

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

       3. *-commutative 67.3%

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

       4. associate-*r/ 67.7%

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

       5. metadata-eval 67.7%

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

   12. Simplified 67.7%

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

   if -2.2000000000000002e155 < l < -7.7999999999999999e-191 or 2.4e-102 < l < 1.70000000000000009e129

   1. Initial program 63.5%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in U around 0 66.9%

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

      1. associate-*r* 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-/l* 67.0%

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

      4. *-commutative 67.0%

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

      5. fma-def 67.0%

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

      6. associate-/l* 68.5%

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

   6. Simplified 68.5%

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

   if -7.7999999999999999e-191 < l < 2.4e-102

   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 49.3%

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

   4. Taylor expanded in U* around 0 53.7%

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

      1. associate-*r* 60.6%

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

      2. +-commutative 60.6%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in Om around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. unpow2 60.6%

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

   9. Simplified 60.6%

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

   if 1.70000000000000009e129 < l

   1. Initial program 10.5%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in U around 0 23.9%

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

      1. associate-*r* 23.9%

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

      2. *-commutative 23.9%

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

      3. associate-/l* 23.9%

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

      4. *-commutative 23.9%

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

      5. fma-def 23.9%

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

      6. associate-/l* 23.9%

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

   6. Simplified 23.9%

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

   7. Taylor expanded in t around 0 35.6%

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

      1. associate-/l* 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*l/ 35.6%

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

      4. *-commutative 35.6%

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

      5. fma-udef 35.6%

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

      6. *-commutative 35.6%

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

   9. Simplified 35.6%

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

   10. Taylor expanded in l around 0 72.4%

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

       1. associate-*l* 72.5%

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

       2. associate-/l* 80.2%

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

       3. *-commutative 80.2%

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

       4. sub-neg 80.2%

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

       5. *-commutative 80.2%

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

       6. associate-*r/ 80.0%

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

       7. metadata-eval 80.0%

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

   12. Simplified 80.0%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 2: 64.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 2e-145)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l (/ l Om)))) t_1)))
       (sqrt
        (* (* 2.0 n) (/ l (/ Om (* U (* l (+ -2.0 (* U* (/ n Om)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-145) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)));
	} else {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-145) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)))
	tmp = 0
	if t_3 <= 2e-145:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)))
	else:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 2e-145)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_1)));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	tmp = 0.0;
	if (t_3 <= 2e-145)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)));
	else
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-145], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999983e-145

   1. Initial program 19.7%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in U around 0 49.1%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 71.8%

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

      1. associate-/l* 73.5%

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

      2. associate-/r/ 73.5%

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

   3. Applied egg-rr 73.5%

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

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

   1. Initial program 0.0%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in U around 0 38.8%

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

      1. associate-*r* 38.8%

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

      2. *-commutative 38.8%

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

      3. associate-/l* 45.1%

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

      4. *-commutative 45.1%

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

      5. fma-def 45.1%

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

      6. associate-/l* 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in t around 0 47.0%

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

      1. associate-/l* 51.2%

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

      2. *-commutative 51.2%

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

      3. associate-*l/ 51.1%

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

      4. *-commutative 51.1%

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

      5. fma-udef 51.1%

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

      6. *-commutative 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in l around 0 55.4%

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

       1. associate-*r* 55.4%

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

       2. sub-neg 55.4%

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

       3. *-commutative 55.4%

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

       4. associate-*r/ 57.8%

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

       5. metadata-eval 57.8%

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

   12. Simplified 57.8%

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

4. Final simplification 66.7%

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

Alternative 3: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -9e+179)
   (*
    (sqrt 2.0)
    (* (sqrt (* n (* U (- (* (/ n (* Om Om)) (- U* U)) (/ 2.0 Om))))) (- l)))
   (if (<= l 2.5e+127)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ l (/ Om (fma l -2.0 (/ n (/ Om (* l U*))))))))))
     (* (sqrt 2.0) (* l (sqrt (/ n (/ Om (* U (+ -2.0 (* U* (/ n Om))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -9e+179) {
		tmp = sqrt(2.0) * (sqrt((n * (U * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om))))) * -l);
	} else if (l <= 2.5e+127) {
		tmp = sqrt(((2.0 * n) * (U * (t + (l / (Om / fma(l, -2.0, (n / (Om / (l * U_42_))))))))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * (-2.0 + (U_42_ * (n / Om))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -9e+179)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) - Float64(2.0 / Om))))) * Float64(-l)));
	elseif (l <= 2.5e+127)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(l / Float64(Om / fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_))))))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -9e+179], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e+127], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(l / N[(Om / N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.0000000000000005e179

   1. Initial program 9.8%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in l around inf 35.5%

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

      1. unpow2 35.5%

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

      2. *-commutative 35.5%

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

      3. associate-/l* 35.5%

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

      4. unpow2 35.5%

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

      5. associate-*r/ 35.5%

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

      6. metadata-eval 35.5%

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

   6. Simplified 35.5%

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

   7. Taylor expanded in l around -inf 67.5%

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

      1. mul-1-neg 67.5%

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

      2. *-commutative 67.5%

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

      3. associate-*l* 67.4%

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

      4. *-commutative 67.4%

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

      5. associate-*l/ 67.4%

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

      6. unpow2 67.4%

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

      7. *-commutative 67.4%

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

      8. associate-*r/ 67.4%

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

   9. Simplified 67.4%

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

   if -9.0000000000000005e179 < l < 2.5000000000000002e127

   1. Initial program 59.1%

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

   3. Simplified 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 U around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. *-commutative 63.0%

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

      3. associate-/l* 63.0%

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

      4. *-commutative 63.0%

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

      5. fma-def 63.0%

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

      6. associate-/l* 64.0%

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

   6. Simplified 64.0%

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

   if 2.5000000000000002e127 < l

   1. Initial program 10.5%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in U around 0 23.9%

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

      1. associate-*r* 23.9%

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

      2. *-commutative 23.9%

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

      3. associate-/l* 23.9%

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

      4. *-commutative 23.9%

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

      5. fma-def 23.9%

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

      6. associate-/l* 23.9%

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

   6. Simplified 23.9%

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

   7. Taylor expanded in t around 0 35.6%

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

      1. associate-/l* 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*l/ 35.6%

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

      4. *-commutative 35.6%

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

      5. fma-udef 35.6%

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

      6. *-commutative 35.6%

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

   9. Simplified 35.6%

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

   10. Taylor expanded in l around 0 72.4%

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

       1. associate-*l* 72.5%

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

       2. associate-/l* 80.2%

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

       3. *-commutative 80.2%

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

       4. sub-neg 80.2%

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

       5. *-commutative 80.2%

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

       6. associate-*r/ 80.0%

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

       7. metadata-eval 80.0%

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

   12. Simplified 80.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 4: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 + U* \cdot \frac{n}{Om}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot t_1\right)}}}\\ \mathbf{elif}\;\ell \leq 1.56 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot t_1}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ -2.0 (* U* (/ n Om)))))
   (if (<= l -3.3e+71)
     (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l t_1))))))
     (if (<= l 1.56e+127)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
       (* (sqrt 2.0) (* l (sqrt (/ n (/ Om (* U t_1))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = -2.0 + (U_42_ * (n / Om));
	double tmp;
	if (l <= -3.3e+71) {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * t_1))))));
	} else if (l <= 1.56e+127) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-2.0d0) + (u_42 * (n / om))
    if (l <= (-3.3d+71)) then
        tmp = sqrt(((2.0d0 * n) * (l / (om / (u * (l * t_1))))))
    else if (l <= 1.56d+127) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt(2.0d0) * (l * sqrt((n / (om / (u * t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = -2.0 + (U_42_ * (n / Om));
	double tmp;
	if (l <= -3.3e+71) {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * t_1))))));
	} else if (l <= 1.56e+127) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((n / (Om / (U * t_1)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = -2.0 + (U_42_ * (n / Om))
	tmp = 0
	if l <= -3.3e+71:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * t_1))))))
	elif l <= 1.56e+127:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((n / (Om / (U * t_1)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))
	tmp = 0.0
	if (l <= -3.3e+71)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * t_1))))));
	elseif (l <= 1.56e+127)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = -2.0 + (U_42_ * (n / Om));
	tmp = 0.0;
	if (l <= -3.3e+71)
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * t_1))))));
	elseif (l <= 1.56e+127)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.3e+71], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.56e+127], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2999999999999998e71

   1. Initial program 31.2%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in U around 0 45.6%

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

      1. associate-*r* 45.6%

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

      2. *-commutative 45.6%

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

      3. associate-/l* 54.0%

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

      4. *-commutative 54.0%

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

      5. fma-def 54.0%

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

      6. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in t around 0 45.7%

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

      1. associate-/l* 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*l/ 54.1%

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

      4. *-commutative 54.1%

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

      5. fma-udef 54.1%

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

      6. *-commutative 54.1%

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

   9. Simplified 54.1%

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

   10. Taylor expanded in l around 0 60.6%

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

       1. associate-*r* 60.6%

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

       2. sub-neg 60.6%

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

       3. *-commutative 60.6%

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

       4. associate-*r/ 63.0%

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

       5. metadata-eval 63.0%

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

   12. Simplified 63.0%

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

   if -3.2999999999999998e71 < l < 1.55999999999999997e127

   1. Initial program 60.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in 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 1.55999999999999997e127 < l

   1. Initial program 10.5%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in U around 0 23.9%

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

      1. associate-*r* 23.9%

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

      2. *-commutative 23.9%

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

      3. associate-/l* 23.9%

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

      4. *-commutative 23.9%

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

      5. fma-def 23.9%

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

      6. associate-/l* 23.9%

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

   6. Simplified 23.9%

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

   7. Taylor expanded in t around 0 35.6%

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

      1. associate-/l* 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*l/ 35.6%

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

      4. *-commutative 35.6%

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

      5. fma-udef 35.6%

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

      6. *-commutative 35.6%

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

   9. Simplified 35.6%

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

   10. Taylor expanded in l around 0 72.4%

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

       1. associate-*l* 72.5%

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

       2. associate-/l* 80.2%

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

       3. *-commutative 80.2%

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

       4. sub-neg 80.2%

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

       5. *-commutative 80.2%

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

       6. associate-*r/ 80.0%

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

       7. metadata-eval 80.0%

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

   12. Simplified 80.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{elif}\;\ell \leq 1.56 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)}}}\right)\\ \end{array} \]

Alternative 5: 55.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-189} \lor \neg \left(\ell \leq 2.05 \cdot 10^{-101}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -4.4e+154)
   (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l (+ -2.0 (* U* (/ n Om)))))))))
   (if (or (<= l -1.1e-189) (not (<= l 2.05e-101)))
     (sqrt
      (* (* 2.0 n) (* U (+ t (/ (* l l) (/ Om (+ -2.0 (/ n (/ Om U*)))))))))
     (sqrt (* 2.0 (* U (* n (+ t (/ (* (* l l) -2.0) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -4.4e+154) {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	} else if ((l <= -1.1e-189) || !(l <= 2.05e-101)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -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 <= (-4.4d+154)) then
        tmp = sqrt(((2.0d0 * n) * (l / (om / (u * (l * ((-2.0d0) + (u_42 * (n / om)))))))))
    else if ((l <= (-1.1d-189)) .or. (.not. (l <= 2.05d-101))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * l) / (om / ((-2.0d0) + (n / (om / u_42)))))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) * (-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 <= -4.4e+154) {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	} else if ((l <= -1.1e-189) || !(l <= 2.05e-101)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -4.4e+154:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))))
	elif (l <= -1.1e-189) or not (l <= 2.05e-101):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -4.4e+154)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	elseif ((l <= -1.1e-189) || !(l <= 2.05e-101))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * l) / Float64(Om / Float64(-2.0 + Float64(n / Float64(Om / U_42_)))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) * -2.0) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -4.4e+154)
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	elseif ((l <= -1.1e-189) || ~((l <= 2.05e-101)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -4.4e+154], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, -1.1e-189], N[Not[LessEqual[l, 2.05e-101]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] / N[(Om / N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.4000000000000002e154

   1. Initial program 20.9%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in U around 0 43.7%

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

      1. associate-*r* 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 53.7%

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

      4. *-commutative 53.7%

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

      5. fma-def 53.7%

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

      6. associate-/l* 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in t around 0 50.4%

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

      1. associate-/l* 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*l/ 60.4%

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

      4. *-commutative 60.4%

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

      5. fma-udef 60.4%

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

      6. *-commutative 60.4%

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

   9. Simplified 60.4%

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

   10. Taylor expanded in l around 0 67.3%

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

       1. associate-*r* 67.3%

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

       2. sub-neg 67.3%

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

       3. *-commutative 67.3%

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

       4. associate-*r/ 67.7%

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

       5. metadata-eval 67.7%

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

   12. Simplified 67.7%

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

   if -4.4000000000000002e154 < l < -1.1000000000000001e-189 or 2.05000000000000013e-101 < l

   1. Initial program 55.0%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in U around 0 60.1%

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

   5. Taylor expanded in l around 0 59.3%

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

      1. associate-/l* 58.5%

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

      2. unpow2 58.5%

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

      3. sub-neg 58.5%

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

      4. associate-/l* 60.7%

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

      5. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

   if -1.1000000000000001e-189 < l < 2.05000000000000013e-101

   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 49.3%

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

   4. Taylor expanded in U* around 0 53.7%

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

      1. associate-*r* 60.6%

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

      2. +-commutative 60.6%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in Om around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. unpow2 60.6%

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

   9. Simplified 60.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-189} \lor \neg \left(\ell \leq 2.05 \cdot 10^{-101}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.26 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (* 2.0 n) (/ l (/ Om (* U (* l (+ -2.0 (* U* (/ n Om)))))))))))
   (if (<= l -3.5e+51)
     t_1
     (if (<= l -1.26e-189)
       (sqrt (* (* 2.0 n) (* U (+ t (/ (* n (* (* l l) U*)) (* Om Om))))))
       (if (<= l 1.62e-24)
         (sqrt (* 2.0 (* U (* n (+ t (/ (* (* l l) -2.0) Om))))))
         t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	double tmp;
	if (l <= -3.5e+51) {
		tmp = t_1;
	} else if (l <= -1.26e-189) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))));
	} else if (l <= 1.62e-24) {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * (l / (om / (u * (l * ((-2.0d0) + (u_42 * (n / om)))))))))
    if (l <= (-3.5d+51)) then
        tmp = t_1
    else if (l <= (-1.26d-189)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((n * ((l * l) * u_42)) / (om * om))))))
    else if (l <= 1.62d-24) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) * (-2.0d0)) / om))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	double tmp;
	if (l <= -3.5e+51) {
		tmp = t_1;
	} else if (l <= -1.26e-189) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))));
	} else if (l <= 1.62e-24) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))))
	tmp = 0
	if l <= -3.5e+51:
		tmp = t_1
	elif l <= -1.26e-189:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))))
	elif l <= 1.62e-24:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))))
	tmp = 0.0
	if (l <= -3.5e+51)
		tmp = t_1;
	elseif (l <= -1.26e-189)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(n * Float64(Float64(l * l) * U_42_)) / Float64(Om * Om))))));
	elseif (l <= 1.62e-24)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) * -2.0) / Om))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	tmp = 0.0;
	if (l <= -3.5e+51)
		tmp = t_1;
	elseif (l <= -1.26e-189)
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))));
	elseif (l <= 1.62e-24)
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.5e+51], t$95$1, If[LessEqual[l, -1.26e-189], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(n * N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.62e-24], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.5e51 or 1.62e-24 < l

   1. Initial program 34.1%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in U around 0 48.2%

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

      1. associate-*r* 48.3%

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

      2. *-commutative 48.3%

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

      3. associate-/l* 52.1%

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

      4. *-commutative 52.1%

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

      5. fma-def 52.1%

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

      6. associate-/l* 54.0%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in t around 0 45.5%

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

      1. associate-/l* 50.1%

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

      2. *-commutative 50.1%

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

      3. associate-*l/ 51.0%

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

      4. *-commutative 51.0%

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

      5. fma-udef 51.0%

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

      6. *-commutative 51.0%

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

   9. Simplified 51.0%

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

   10. Taylor expanded in l around 0 54.0%

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

       1. associate-*r* 54.0%

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

       2. sub-neg 54.0%

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

       3. *-commutative 54.0%

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

       4. associate-*r/ 55.0%

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

       5. metadata-eval 55.0%

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

   12. Simplified 55.0%

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

   if -3.5e51 < l < -1.2600000000000001e-189

   1. Initial program 76.8%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in U around 0 75.5%

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

   5. Taylor expanded in n around inf 72.1%

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

      1. *-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. unpow2 72.1%

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

   7. Simplified 72.1%

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

   if -1.2600000000000001e-189 < l < 1.62e-24

   1. Initial program 51.5%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in U* around 0 51.5%

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

      1. associate-*r* 56.7%

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

      2. +-commutative 56.7%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in Om around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. unpow2 56.4%

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

   9. Simplified 56.4%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{elif}\;\ell \leq -1.26 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \]

Alternative 7: 55.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -4.8e+71)
   (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l (+ -2.0 (* U* (/ n Om)))))))))
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -4.8e+71) {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-4.8d+71)) then
        tmp = sqrt(((2.0d0 * n) * (l / (om / (u * (l * ((-2.0d0) + (u_42 * (n / om)))))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -4.8e+71) {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -4.8e+71:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -4.8e+71)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -4.8e+71)
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -4.8e+71], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.79999999999999961e71

   1. Initial program 31.2%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in U around 0 45.6%

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

      1. associate-*r* 45.6%

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

      2. *-commutative 45.6%

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

      3. associate-/l* 54.0%

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

      4. *-commutative 54.0%

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

      5. fma-def 54.0%

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

      6. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in t around 0 45.7%

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

      1. associate-/l* 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*l/ 54.1%

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

      4. *-commutative 54.1%

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

      5. fma-udef 54.1%

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

      6. *-commutative 54.1%

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

   9. Simplified 54.1%

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

   10. Taylor expanded in l around 0 60.6%

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

       1. associate-*r* 60.6%

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

       2. sub-neg 60.6%

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

       3. *-commutative 60.6%

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

       4. associate-*r/ 63.0%

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

       5. metadata-eval 63.0%

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

   12. Simplified 63.0%

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

   if -4.79999999999999961e71 < l

   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 55.2%

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

   4. Taylor expanded in U around 0 59.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 8: 48.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.8 \cdot 10^{-102} \lor \neg \left(Om \leq 4 \cdot 10^{-57}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.8e-102) (not (<= Om 4e-57)))
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt (* -2.0 (/ (* (* n (* l l)) (* U (- 2.0 (/ n (/ Om U*))))) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.8e-102) || !(Om <= 4e-57)) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.8d-102)) .or. (.not. (om <= 4d-57))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * (((n * (l * l)) * (u * (2.0d0 - (n / (om / u_42))))) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.8e-102) || !(Om <= 4e-57)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.8e-102) or not (Om <= 4e-57):
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.8e-102) || !(Om <= 4e-57))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(n * Float64(l * l)) * Float64(U * Float64(2.0 - Float64(n / Float64(Om / U_42_))))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.8e-102) || ~((Om <= 4e-57)))
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.8e-102], N[Not[LessEqual[Om, 4e-57]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.8e-102 or 3.99999999999999982e-57 < Om

   1. Initial program 55.7%

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

      1. associate-*l* 56.0%

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

      2. sub-neg 56.0%

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

      3. associate-+l- 56.0%

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

      4. sub-neg 56.0%

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

      5. associate-/l* 59.1%

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

      6. remove-double-neg 59.1%

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

      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 51.5%

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

      1. unpow2 51.5%

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

      2. associate-*r/ 54.2%

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

   6. Simplified 54.2%

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

   if -1.8e-102 < Om < 3.99999999999999982e-57

   1. Initial program 34.4%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in U around 0 57.3%

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

   5. Taylor expanded in l around -inf 49.3%

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

      1. associate-*r* 48.4%

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

      2. unpow2 48.4%

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

      3. *-commutative 48.4%

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

      4. mul-1-neg 48.4%

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

      5. unsub-neg 48.4%

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

      6. associate-/l* 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.8 \cdot 10^{-102} \lor \neg \left(Om \leq 4 \cdot 10^{-57}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \end{array} \]

Alternative 9: 48.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -0.0032 \lor \neg \left(U* \leq 5.8 \cdot 10^{-39}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -0.0032) (not (<= U* 5.8e-39)))
   (sqrt (* (* 2.0 n) (* U (+ t (/ (* n (* (* l l) U*)) (* Om Om))))))
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -0.0032) || !(U_42_ <= 5.8e-39)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-0.0032d0)) .or. (.not. (u_42 <= 5.8d-39))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((n * ((l * l) * u_42)) / (om * om))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -0.0032) || !(U_42_ <= 5.8e-39)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -0.0032) or not (U_42_ <= 5.8e-39):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -0.0032) || !(U_42_ <= 5.8e-39))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(n * Float64(Float64(l * l) * U_42_)) / Float64(Om * Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -0.0032) || ~((U_42_ <= 5.8e-39)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * ((l * l) * U_42_)) / (Om * Om))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -0.0032], N[Not[LessEqual[U$42$, 5.8e-39]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(n * N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -0.00320000000000000015 or 5.79999999999999975e-39 < U*

   1. Initial program 47.2%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in U around 0 57.7%

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

   5. Taylor expanded in n around inf 52.6%

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

      1. *-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

   7. Simplified 52.6%

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

   if -0.00320000000000000015 < U* < 5.79999999999999975e-39

   1. Initial program 55.5%

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

      1. associate-*l* 53.6%

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

      2. sub-neg 53.6%

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

      3. associate-+l- 53.6%

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

      4. sub-neg 53.6%

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

      5. associate-/l* 57.7%

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

      6. remove-double-neg 57.7%

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

      7. associate-*l* 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in Om around inf 55.7%

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

      1. unpow2 55.7%

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

      2. associate-*r/ 59.8%

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

   6. Simplified 59.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -0.0032 \lor \neg \left(U* \leq 5.8 \cdot 10^{-39}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10: 48.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.2 \cdot 10^{-192} \lor \neg \left(Om \leq 1.5 \cdot 10^{-62}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot U*\right)}{Om}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -2.2e-192) (not (<= Om 1.5e-62)))
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt (* (* 2.0 n) (/ l (/ Om (/ (* (* n l) (* U U*)) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.2e-192) || !(Om <= 1.5e-62)) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(((2.0 * n) * (l / (Om / (((n * l) * (U * U_42_)) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-2.2d-192)) .or. (.not. (om <= 1.5d-62))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt(((2.0d0 * n) * (l / (om / (((n * l) * (u * u_42)) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.2e-192) || !(Om <= 1.5e-62)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (((n * l) * (U * U_42_)) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -2.2e-192) or not (Om <= 1.5e-62):
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (((n * l) * (U * U_42_)) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -2.2e-192) || !(Om <= 1.5e-62))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(Float64(Float64(n * l) * Float64(U * U_42_)) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -2.2e-192) || ~((Om <= 1.5e-62)))
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt(((2.0 * n) * (l / (Om / (((n * l) * (U * U_42_)) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -2.2e-192], N[Not[LessEqual[Om, 1.5e-62]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(N[(N[(n * l), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.20000000000000006e-192 or 1.5000000000000001e-62 < Om

   1. Initial program 54.5%

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

      1. associate-*l* 55.7%

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

      2. sub-neg 55.7%

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

      3. associate-+l- 55.7%

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

      4. sub-neg 55.7%

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

      5. associate-/l* 58.7%

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

      6. remove-double-neg 58.7%

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

      7. associate-*l* 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in Om around inf 51.0%

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

      1. unpow2 51.0%

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

      2. associate-*r/ 53.5%

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

   6. Simplified 53.5%

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

   if -2.20000000000000006e-192 < Om < 1.5000000000000001e-62

   1. Initial program 34.1%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in U around 0 53.7%

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

      1. associate-*r* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-/l* 50.1%

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

      4. *-commutative 50.1%

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

      5. fma-def 50.1%

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

      6. associate-/l* 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in t around 0 51.9%

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

      1. associate-/l* 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*l/ 50.3%

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

      4. *-commutative 50.3%

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

      5. fma-udef 50.3%

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

      6. *-commutative 50.3%

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

   9. Simplified 50.3%

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

   10. Taylor expanded in U* around inf 44.3%

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

       1. associate-*r* 46.2%

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

       2. *-commutative 46.2%

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

   12. Simplified 46.2%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.2 \cdot 10^{-192} \lor \neg \left(Om \leq 1.5 \cdot 10^{-62}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot U*\right)}{Om}}}}\\ \end{array} \]

Alternative 11: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+48} \lor \neg \left(\ell \leq 1.6 \cdot 10^{-6}\right):\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -7.6e+48) (not (<= l 1.6e-6)))
   (pow (* (* 2.0 n) (* -2.0 (* U (/ (* l l) Om)))) 0.5)
   (pow (* (* 2.0 n) (* U t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -7.6e+48) || !(l <= 1.6e-6)) {
		tmp = pow(((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))), 0.5);
	} else {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((l <= (-7.6d+48)) .or. (.not. (l <= 1.6d-6))) then
        tmp = ((2.0d0 * n) * ((-2.0d0) * (u * ((l * l) / om)))) ** 0.5d0
    else
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -7.6e+48) || !(l <= 1.6e-6)) {
		tmp = Math.pow(((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -7.6e+48) or not (l <= 1.6e-6):
		tmp = math.pow(((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))), 0.5)
	else:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -7.6e+48) || !(l <= 1.6e-6))
		tmp = Float64(Float64(2.0 * n) * Float64(-2.0 * Float64(U * Float64(Float64(l * l) / Om)))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -7.6e+48) || ~((l <= 1.6e-6)))
		tmp = ((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))) ^ 0.5;
	else
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -7.6e+48], N[Not[LessEqual[l, 1.6e-6]], $MachinePrecision]], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.60000000000000001e48 or 1.5999999999999999e-6 < 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 54.4%

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

   4. Taylor expanded in l around inf 41.0%

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

      1. unpow2 41.0%

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

      2. *-commutative 41.0%

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

      3. associate-/l* 41.1%

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

      4. unpow2 41.1%

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

      5. associate-*r/ 41.1%

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

      6. metadata-eval 41.1%

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

   6. Simplified 41.1%

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

   7. Taylor expanded in n around 0 23.6%

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

      1. associate-/l* 23.9%

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

      2. unpow2 23.9%

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

   9. Simplified 23.9%

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

       1. pow1/2 38.4%

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

       2. associate-/r/ 41.0%

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

   11. Applied egg-rr 41.0%

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

   if -7.60000000000000001e48 < l < 1.5999999999999999e-6

   1. Initial program 62.1%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in t around inf 52.3%

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

      1. pow1/2 53.1%

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

      2. *-commutative 53.1%

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

   6. Applied egg-rr 53.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+48} \lor \neg \left(\ell \leq 1.6 \cdot 10^{-6}\right):\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 12: 43.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+48} \lor \neg \left(\ell \leq 1.42 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot -2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -4e+48) (not (<= l 1.42e-5)))
   (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l -2.0))))))
   (pow (* (* 2.0 n) (* U t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -4e+48) || !(l <= 1.42e-5)) {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * -2.0))))));
	} else {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((l <= (-4d+48)) .or. (.not. (l <= 1.42d-5))) then
        tmp = sqrt(((2.0d0 * n) * (l / (om / (u * (l * (-2.0d0)))))))
    else
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -4e+48) || !(l <= 1.42e-5)) {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * -2.0))))));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -4e+48) or not (l <= 1.42e-5):
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * -2.0))))))
	else:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -4e+48) || !(l <= 1.42e-5))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * -2.0))))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -4e+48) || ~((l <= 1.42e-5)))
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * -2.0))))));
	else
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -4e+48], N[Not[LessEqual[l, 1.42e-5]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.00000000000000018e48 or 1.42e-5 < 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 54.4%

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

   4. Taylor expanded in U around 0 46.6%

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

      1. associate-*r* 46.7%

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

      2. *-commutative 46.7%

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

      3. associate-/l* 50.6%

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

      4. *-commutative 50.6%

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

      5. fma-def 50.6%

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

      6. associate-/l* 52.5%

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

   6. Simplified 52.5%

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

   7. Taylor expanded in t around 0 44.7%

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

      1. associate-/l* 49.5%

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

      2. *-commutative 49.5%

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

      3. associate-*l/ 50.4%

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

      4. *-commutative 50.4%

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

      5. fma-udef 50.4%

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

      6. *-commutative 50.4%

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

   9. Simplified 50.4%

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

   10. Taylor expanded in U* around 0 34.3%

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

       1. associate-*r* 34.3%

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

       2. *-commutative 34.3%

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

   12. Simplified 34.3%

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

   if -4.00000000000000018e48 < l < 1.42e-5

   1. Initial program 62.1%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in t around inf 52.3%

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

      1. pow1/2 53.1%

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

      2. *-commutative 53.1%

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

   6. Applied egg-rr 53.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+48} \lor \neg \left(\ell \leq 1.42 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot -2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 13: 38.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 0.022:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.02e+49)
   (sqrt (/ (* -4.0 (* n (* U (* l l)))) Om))
   (if (<= l 0.022)
     (pow (* (* 2.0 n) (* U t)) 0.5)
     (sqrt (* (* 2.0 n) (* -2.0 (/ (* l l) (/ Om U))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.02e+49) {
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	} else if (l <= 0.022) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / 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 <= (-1.02d+49)) then
        tmp = sqrt((((-4.0d0) * (n * (u * (l * l)))) / om))
    else if (l <= 0.022d0) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * n) * ((-2.0d0) * ((l * l) / (om / 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 <= -1.02e+49) {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	} else if (l <= 0.022) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -1.02e+49:
		tmp = math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om))
	elif l <= 0.022:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.02e+49)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l * l)))) / Om));
	elseif (l <= 0.022)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(-2.0 * Float64(Float64(l * l) / Float64(Om / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -1.02e+49)
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	elseif (l <= 0.022)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = sqrt(((2.0 * n) * (-2.0 * ((l * l) / (Om / U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.02e+49], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 0.022], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(-2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.02e49

   1. Initial program 30.9%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in l around inf 43.7%

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

      1. unpow2 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 43.7%

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

      4. unpow2 43.7%

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

      5. associate-*r/ 43.7%

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

      6. metadata-eval 43.7%

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

   6. Simplified 43.7%

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

   7. Taylor expanded in n around 0 27.6%

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

      1. associate-/l* 25.7%

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

      2. unpow2 25.7%

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

   9. Simplified 25.7%

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

   10. Taylor expanded in n around 0 27.6%

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

       1. associate-*r/ 27.6%

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

       2. *-commutative 27.6%

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

       3. unpow2 27.6%

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

   12. Simplified 27.6%

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

   if -1.02e49 < l < 0.021999999999999999

   1. Initial program 61.8%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in t around inf 52.1%

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

      1. pow1/2 52.9%

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

      2. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

   if 0.021999999999999999 < l

   1. Initial program 33.7%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in l around inf 38.9%

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

      1. unpow2 38.9%

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

      2. *-commutative 38.9%

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

      3. associate-/l* 38.9%

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

      4. unpow2 38.9%

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

      5. associate-*r/ 38.9%

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

      6. metadata-eval 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in n around 0 19.8%

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

      1. associate-/l* 22.3%

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

      2. unpow2 22.3%

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

   9. Simplified 22.3%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 0.022:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \end{array} \]

Alternative 14: 45.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+76}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -3e+76)
   (pow (* (* 2.0 n) (* -2.0 (* U (/ (* l l) Om)))) 0.5)
   (sqrt (* 2.0 (* U (* n (+ t (/ (* (* l l) -2.0) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -3e+76) {
		tmp = pow(((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -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 <= (-3d+76)) then
        tmp = ((2.0d0 * n) * ((-2.0d0) * (u * ((l * l) / om)))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) * (-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 <= -3e+76) {
		tmp = Math.pow(((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -3e+76:
		tmp = math.pow(((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -3e+76)
		tmp = Float64(Float64(2.0 * n) * Float64(-2.0 * Float64(U * Float64(Float64(l * l) / Om)))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) * -2.0) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -3e+76)
		tmp = ((2.0 * n) * (-2.0 * (U * ((l * l) / Om)))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) * -2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -3e+76], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.9999999999999998e76

   1. Initial program 30.3%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in l around inf 44.7%

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

      1. unpow2 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-/l* 44.8%

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

      4. unpow2 44.8%

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

      5. associate-*r/ 44.8%

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

      6. metadata-eval 44.8%

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

   6. Simplified 44.8%

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

   7. Taylor expanded in n around 0 26.4%

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

      1. associate-/l* 24.2%

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

      2. unpow2 24.2%

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

   9. Simplified 24.2%

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

       1. pow1/2 40.9%

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

       2. associate-/r/ 45.4%

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

   11. Applied egg-rr 45.4%

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

   if -2.9999999999999998e76 < l

   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 55.2%

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

   4. Taylor expanded in U* around 0 45.5%

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

      1. associate-*r* 48.2%

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

      2. +-commutative 48.2%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in Om around inf 51.7%

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

      1. associate-*r/ 51.7%

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

      2. unpow2 51.7%

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

   9. Simplified 51.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+76}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 15: 46.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)))))))
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)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. associate-*l* 51.3%

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

   2. sub-neg 51.3%

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

   3. associate-+l- 51.3%

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

   4. sub-neg 51.3%

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

   5. associate-/l* 53.7%

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

   6. remove-double-neg 53.7%

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

   7. associate-*l* 52.0%

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

3. Simplified 52.0%

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

4. Taylor expanded in Om around inf 45.9%

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

   1. unpow2 45.9%

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

   2. associate-*r/ 47.9%

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

6. Simplified 47.9%

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

7. Final simplification 47.9%

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

Alternative 16: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -6.5e+48)
   (sqrt (/ (* -4.0 (* n (* U (* l l)))) Om))
   (pow (* (* 2.0 n) (* U t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -6.5e+48) {
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	} else {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-6.5d+48)) then
        tmp = sqrt((((-4.0d0) * (n * (u * (l * l)))) / om))
    else
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -6.5e+48) {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -6.5e+48:
		tmp = math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om))
	else:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -6.5e+48)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l * l)))) / Om));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -6.5e+48)
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	else
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -6.5e+48], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.49999999999999972e48

   1. Initial program 30.9%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in l around inf 43.7%

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

      1. unpow2 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 43.7%

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

      4. unpow2 43.7%

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

      5. associate-*r/ 43.7%

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

      6. metadata-eval 43.7%

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

   6. Simplified 43.7%

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

   7. Taylor expanded in n around 0 27.6%

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

      1. associate-/l* 25.7%

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

      2. unpow2 25.7%

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

   9. Simplified 25.7%

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

   10. Taylor expanded in n around 0 27.6%

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

       1. associate-*r/ 27.6%

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

       2. *-commutative 27.6%

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

       3. unpow2 27.6%

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

   12. Simplified 27.6%

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

   if -6.49999999999999972e48 < l

   1. Initial program 55.0%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in t around inf 41.9%

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

      1. pow1/2 43.0%

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

      2. *-commutative 43.0%

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

   6. Applied egg-rr 43.0%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 17: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = ((2.0d0 * n) * (u * t)) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow(((2.0 * n) * (U * t)), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 56.5%

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

4. Taylor expanded in t around inf 35.9%

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

   1. pow1/2 37.9%

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

   2. *-commutative 37.9%

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

6. Applied egg-rr 37.9%

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

7. Final simplification 37.9%

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

Alternative 18: 34.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 56.5%

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

4. Taylor expanded in t around inf 35.9%

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

5. Final simplification 35.9%

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

Reproduce

herbie shell --seed 2023192 
(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*))))))