Toniolo and Linder, Equation (13)

Percentage Accurate: 50.6% → 65.9%

Time: 24.7s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.6% 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: 65.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t_2\right) \cdot \left(U* - U\right)\\ t_4 := \left(2 \cdot n\right) \cdot U\\ t_5 := \sqrt{t_4 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_3\right)}\\ \mathbf{if}\;t_5 \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(n, t_2 \cdot \left(U* - U\right), \mathsf{fma}\left(t_1, -2, t\right)\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\sqrt{t_4 \cdot \left(\left(t - 2 \cdot t_1\right) + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (* (* 2.0 n) U))
        (t_5 (sqrt (* t_4 (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_5 5e-159)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (fma n (* t_2 (- U* U)) (fma t_1 -2.0 t)))))
     (if (<= t_5 INFINITY)
       (sqrt (* t_4 (+ (- t (* 2.0 t_1)) t_3)))
       (*
        (sqrt
         (* U (* n (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 Om))))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = (2.0 * n) * U;
	double t_5 = sqrt((t_4 * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_5 <= 5e-159) {
		tmp = sqrt((2.0 * n)) * sqrt((U * fma(n, (t_2 * (U_42_ - U)), fma(t_1, -2.0, t))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_4 * ((t - (2.0 * t_1)) + t_3)));
	} else {
		tmp = sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om)))))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = Float64(Float64(2.0 * n) * U)
	t_5 = sqrt(Float64(t_4 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_5 <= 5e-159)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * fma(n, Float64(t_2 * Float64(U_42_ - U)), fma(t_1, -2.0, t)))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(t_4 * Float64(Float64(t - Float64(2.0 * t_1)) + t_3)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om)))))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t$95$4 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 5e-159], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(t$95$4 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000032e-159

   1. Initial program 8.3%

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

   3. Simplified 31.4%

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

      1. pow1/2 31.4%

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

      2. associate-*r* 31.8%

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

      3. unpow-prod-down 35.3%

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

      4. pow1/2 35.3%

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

      5. pow1/2 35.3%

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

      6. associate-*r/ 35.3%

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

      7. pow2 35.3%

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

   5. Applied egg-rr 35.3%

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

      1. *-commutative 35.3%

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

      2. *-commutative 35.3%

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

   7. Simplified 35.3%

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

      1. unpow2 35.3%

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

      2. associate-*l/ 35.3%

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

   9. Applied egg-rr 35.3%

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

   if 5.00000000000000032e-159 < (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 67.9%

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

      1. associate-*l/ 72.9%

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

   3. Applied egg-rr 72.9%

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

   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 15.8%

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

   4. Taylor expanded in l around inf 29.7%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + t_1 \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot t_1\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_4 0.0)
     (sqrt (* 2.0 (* n (* U (+ t (* t_1 -2.0))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (*
        (sqrt
         (* U (* n (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 Om))))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om)))))) * (l * sqrt(2.0));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.sqrt((U * (n * (((n * (U_42_ - U)) / Math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om)))))) * (l * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.sqrt((U * (n * (((n * (U_42_ - U)) / math.pow(Om, 2.0)) + (2.0 * (-1.0 / Om)))))) * (l * math.sqrt(2.0))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(t_1 * -2.0))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om)))))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = sqrt((U * (n * (((n * (U_42_ - U)) / (Om ^ 2.0)) + (2.0 * (-1.0 / Om)))))) * (l * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.0%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in n around 0 32.4%

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

      1. unpow2 28.7%

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

      2. associate-*l/ 28.7%

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

   6. Applied egg-rr 35.7%

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

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

   1. Initial program 67.8%

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

      1. associate-*l/ 72.8%

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

   3. Applied egg-rr 72.8%

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

   if +inf.0 < (*.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 9.6%

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

   4. Taylor expanded in l around inf 26.7%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 3: 65.8% accurate, 0.4× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_4 0.0)
     (sqrt (* 2.0 (* n (* U (+ t (* t_1 -2.0))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (*
        (* l (sqrt 2.0))
        (sqrt (* U (* n (- (/ n (/ (pow Om 2.0) (- U* U))) (/ 2.0 Om))))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.0%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in n around 0 32.4%

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

      1. unpow2 28.7%

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

      2. associate-*l/ 28.7%

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

   6. Applied egg-rr 35.7%

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

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

   1. Initial program 67.8%

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

      1. associate-*l/ 72.8%

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

   3. Applied egg-rr 72.8%

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

   if +inf.0 < (*.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 9.6%

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

   4. Taylor expanded in l around inf 26.7%

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

      1. associate-/l* 22.5%

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

      2. associate-*r/ 22.5%

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

      3. metadata-eval 22.5%

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

   6. Simplified 22.5%

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

4. Final simplification 62.1%

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

Alternative 4: 63.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + t_1 \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot t_1\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot {\ell}^{2}}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_4 0.0)
     (sqrt (* 2.0 (* n (* U (+ t (* t_1 -2.0))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (pow (* -4.0 (/ U (/ Om (* n (pow l 2.0))))) 0.5)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = pow((-4.0 * (U / (Om / (n * pow(l, 2.0))))), 0.5);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.pow((-4.0 * (U / (Om / (n * Math.pow(l, 2.0))))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.pow((-4.0 * (U / (Om / (n * math.pow(l, 2.0))))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(t_1 * -2.0))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = Float64(-4.0 * Float64(U / Float64(Om / Float64(n * (l ^ 2.0))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = (-4.0 * (U / (Om / (n * (l ^ 2.0))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U / N[(Om / N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.0%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in n around 0 32.4%

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

      1. unpow2 28.7%

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

      2. associate-*l/ 28.7%

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

   6. Applied egg-rr 35.7%

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

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

   1. Initial program 67.8%

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

      1. associate-*l/ 72.8%

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

   3. Applied egg-rr 72.8%

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

   if +inf.0 < (*.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 9.6%

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

   4. Taylor expanded in n around 0 2.3%

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

   5. Taylor expanded in t around 0 11.8%

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

      1. pow1/2 31.9%

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

      2. associate-*r* 31.9%

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

      3. metadata-eval 31.9%

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

      4. associate-/l* 31.9%

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

      5. *-commutative 31.9%

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

   7. Applied egg-rr 31.9%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U}{\frac{Om}{n \cdot {\ell}^{2}}}\right)}^{0.5}\\ \end{array} \]

Alternative 5: 56.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 2.45 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -7.4e-37)
   (pow (* 2.0 (* (* n U) (fma -2.0 (* l (/ l Om)) t))) 0.5)
   (if (<= U 2.45e+76)
     (sqrt
      (*
       (* 2.0 n)
       (*
        U
        (+
         (+ t (* -2.0 (/ l (/ Om l))))
         (* n (* (pow (/ l Om) 2.0) (- U* U)))))))
     (* (sqrt (* 2.0 U)) (sqrt (* n t))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -7.4e-37) {
		tmp = pow((2.0 * ((n * U) * fma(-2.0, (l * (l / Om)), t))), 0.5);
	} else if (U <= 2.45e+76) {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (pow((l / Om), 2.0) * (U_42_ - U)))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -7.4e-37)
		tmp = Float64(2.0 * Float64(Float64(n * U) * fma(-2.0, Float64(l * Float64(l / Om)), t))) ^ 0.5;
	elseif (U <= 2.45e+76)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -7.4e-37], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 2.45e+76], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -7.4e-37

   1. Initial program 66.6%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in n around 0 61.0%

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

      1. pow1/2 63.2%

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

      2. associate-*r* 68.6%

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

      3. +-commutative 68.6%

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

      4. fma-def 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. unpow2 28.7%

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

      2. associate-*l/ 28.7%

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

   8. Applied egg-rr 73.7%

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

   if -7.4e-37 < U < 2.45000000000000013e76

   1. Initial program 44.8%

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

   3. Simplified 52.7%

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

   if 2.45000000000000013e76 < U

   1. Initial program 59.2%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in l around 0 48.6%

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

      1. pow1/2 48.6%

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

      2. associate-*r* 48.6%

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

      3. unpow-prod-down 70.1%

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

      4. pow1/2 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. unpow1/2 70.1%

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

   8. Simplified 70.1%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 2.45 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 6: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + t_1 \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, t_1, t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= t -2.5e+35)
     (sqrt (* 2.0 (* n (* U (+ t (* t_1 -2.0))))))
     (pow (* 2.0 (* (* n U) (fma -2.0 t_1 t))) 0.5))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (t <= -2.5e+35) {
		tmp = sqrt((2.0 * (n * (U * (t + (t_1 * -2.0))))));
	} else {
		tmp = pow((2.0 * ((n * U) * fma(-2.0, t_1, t))), 0.5);
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (t <= -2.5e+35)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(t_1 * -2.0))))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * fma(-2.0, t_1, t))) ^ 0.5;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e+35], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.50000000000000011e35

   1. Initial program 40.4%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in n around 0 45.6%

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

      1. unpow2 32.3%

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

      2. associate-*l/ 34.1%

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

   6. Applied egg-rr 53.0%

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

   if -2.50000000000000011e35 < t

   1. Initial program 54.8%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in n around 0 43.3%

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

      1. pow1/2 46.4%

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

      2. associate-*r* 51.2%

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

      3. +-commutative 51.2%

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

      4. fma-def 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. unpow2 26.2%

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

      2. associate-*l/ 28.1%

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

   8. Applied egg-rr 55.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7: 49.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 3.5e+75)
   (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))
   (* (sqrt (* 2.0 U)) (sqrt (* n t)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 3.5e+75) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 3.5d+75) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 3.5e+75) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 3.5e+75:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 3.5e+75)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 3.5e+75)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 3.5e+75], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 3.4999999999999998e75

   1. Initial program 50.4%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in n around 0 42.7%

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

      1. unpow2 26.8%

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

      2. associate-*l/ 29.0%

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

   6. Applied egg-rr 48.4%

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

   if 3.4999999999999998e75 < U

   1. Initial program 59.2%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in l around 0 48.6%

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

      1. pow1/2 48.6%

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

      2. associate-*r* 48.6%

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

      3. unpow-prod-down 70.1%

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

      4. pow1/2 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. unpow1/2 70.1%

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

   8. Simplified 70.1%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 8: 47.7% accurate, 2.0× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0)))))))

C

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

Fortran

NOTE: l should be positive before calling this function
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 + ((l * (l / om)) * (-2.0d0)))))))
end function

Java

l = Math.abs(l);
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 + ((l * (l / Om)) * -2.0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 54.8%

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

4. Taylor expanded in n around 0 43.7%

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

   1. unpow2 27.5%

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

   2. associate-*l/ 29.3%

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

6. Applied egg-rr 48.6%

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

7. Final simplification 48.6%

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

Alternative 9: 37.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 6e-55)
   (sqrt (* (* (* 2.0 n) U) t))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 6e-55) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 <= 6d-55) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 6e-55) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 6e-55:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 6e-55)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 6e-55)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 6e-55], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 6.00000000000000033e-55

   1. Initial program 51.2%

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

      1. associate-*l/ 56.3%

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

   3. Applied egg-rr 56.3%

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

   4. Taylor expanded in t around inf 36.2%

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

      1. associate-*r* 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-*r* 39.8%

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

      4. associate-*r* 39.9%

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

      5. *-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. expm1-log1p-u 38.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\right)\right)} \]

      2. expm1-udef 21.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\right)} - 1} \]

      3. *-commutative 21.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot t\right)}\right)} - 1 \]

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 38.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\right)\right)} \]

      2. expm1-log1p 39.9%

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

      3. associate-*r* 41.7%

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

      4. *-commutative 41.7%

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

      5. *-commutative 41.7%

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

      6. *-commutative 41.7%

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

   10. Simplified 41.7%

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

   if 6.00000000000000033e-55 < U*

   1. Initial program 53.0%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in n around 0 37.1%

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

      1. pow1/2 40.8%

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

      2. associate-*r* 41.7%

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

      3. +-commutative 41.7%

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

      4. fma-def 41.7%

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

   6. Applied egg-rr 41.7%

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

   7. Taylor expanded in l around 0 35.2%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 10: 35.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 6.8e-226)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* U (* n t))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 6.8e-226) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 (n <= 6.8d-226) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 6.8e-226) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 6.8e-226:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 6.8e-226)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 6.8e-226)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 6.8e-226], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.80000000000000014e-226

   1. Initial program 50.5%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in l around 0 37.5%

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

   if 6.80000000000000014e-226 < n

   1. Initial program 54.2%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in l around 0 39.8%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 11: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -6e+35) (sqrt (* 2.0 (* n (* U t)))) (sqrt (* 2.0 (* t (* n U))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -6e+35) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-6d+35)) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -6e+35) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -6e+35:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -6e+35)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -6e+35)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -6e+35], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.99999999999999981e35

   1. Initial program 40.4%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in l around 0 46.8%

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

   if -5.99999999999999981e35 < t

   1. Initial program 54.8%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in l around 0 33.0%

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

      1. associate-*r* 37.6%

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

      2. *-commutative 37.6%

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

   6. Simplified 37.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 12: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5e+37) (sqrt (* 2.0 (* n (* U t)))) (sqrt (* (* (* 2.0 n) U) t))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e+37) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-5d+37)) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e+37) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5e+37:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5e+37)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5e+37)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((((2.0 * n) * U) * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5e+37], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.99999999999999989e37

   1. Initial program 40.4%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in l around 0 46.8%

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

   if -4.99999999999999989e37 < t

   1. Initial program 54.8%

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

      1. associate-*l/ 59.1%

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

   3. Applied egg-rr 59.1%

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

   4. Taylor expanded in t around inf 33.0%

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

      1. associate-*r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-*r* 33.9%

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

      4. associate-*r* 33.9%

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

      5. *-commutative 33.9%

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

   6. Simplified 33.9%

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

      1. expm1-log1p-u 33.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\right)\right)} \]

      2. expm1-udef 18.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\right)} - 1} \]

      3. *-commutative 18.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot t\right)}\right)} - 1 \]

   8. Applied egg-rr 18.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 33.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\right)\right)} \]

      2. expm1-log1p 33.9%

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

      3. associate-*r* 37.6%

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

      4. *-commutative 37.6%

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

      5. *-commutative 37.6%

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

      6. *-commutative 37.6%

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

   10. Simplified 37.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \]

Alternative 13: 36.4% accurate, 2.1× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

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

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 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 54.8%

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

4. Taylor expanded in l around 0 35.5%

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

5. Final simplification 35.5%

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

Reproduce

herbie shell --seed 2023309 
(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*))))))