Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 65.5%

Time: 27.5s

Alternatives: 14

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: 49.5% 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: 65.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\\ t_3 := t_2 \cdot \left(U* - U\right)\\ t_4 := \sqrt{t_1 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_3\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{l_m}^{2}}{Om}, t_2 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U*}{\frac{{Om}^{2}}{n}} + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(l_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (* n (pow (/ l_m Om) 2.0)))
        (t_3 (* t_2 (- U* U)))
        (t_4 (sqrt (* t_1 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_3)))))
   (if (<= t_4 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (fma 2.0 (/ (pow l_m 2.0) Om) (* t_2 (- U U*)))))))
     (if (<= t_4 2e+147)
       (sqrt (* t_1 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_3)))
       (*
        (sqrt (* (+ (/ U* (/ (pow Om 2.0) n)) (/ -2.0 Om)) (* n U)))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = n * pow((l_m / Om), 2.0);
	double t_3 = t_2 * (U_42_ - U);
	double t_4 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, (pow(l_m, 2.0) / Om), (t_2 * (U - U_42_))))));
	} else if (t_4 <= 2e+147) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_3)));
	} else {
		tmp = sqrt((((U_42_ / (pow(Om, 2.0) / n)) + (-2.0 / Om)) * (n * U))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	t_3 = Float64(t_2 * Float64(U_42_ - U))
	t_4 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64((l_m ^ 2.0) / Om), Float64(t_2 * Float64(U - U_42_)))))));
	elseif (t_4 <= 2e+147)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_3)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(U_42_ / Float64((Om ^ 2.0) / n)) + Float64(-2.0 / Om)) * Float64(n * U))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] + N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+147], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(U$42$ / N[(N[Power[Om, 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * 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*))))) < 0.0

   1. Initial program 12.2%

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

   3. Simplified 45.0%

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

      1. sqrt-prod 51.8%

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

      2. fma-udef 51.8%

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

      3. associate-*l/ 51.8%

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

      4. associate-*r/ 51.8%

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

      5. associate--l- 51.8%

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

      6. associate--l- 51.8%

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

   5. Applied egg-rr 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*r* 51.8%

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

   7. Simplified 51.8%

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

   if 0.0 < (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*))))) < 2e147

   1. Initial program 95.9%

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

      1. associate-*l/ 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 2e147 < (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 17.9%

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

   3. Simplified 26.4%

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

   4. Taylor expanded in l around inf 25.4%

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

      1. associate-*r/ 25.4%

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

      2. metadata-eval 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in U* around inf 25.6%

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

      1. *-commutative 25.6%

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

   9. Simplified 25.6%

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

   10. Taylor expanded in l around 0 21.0%

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

       1. associate-*r* 23.7%

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

       2. sub-neg 23.7%

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

       3. associate-*r/ 23.7%

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

       4. metadata-eval 23.7%

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

       5. associate-/l* 24.6%

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

       6. distribute-neg-frac 24.6%

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

       7. metadata-eval 24.6%

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

   12. Simplified 24.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U*}{\frac{{Om}^{2}}{n}} + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2: 64.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.2%

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

   3. Simplified 44.9%

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

      1. expm1-log1p-u 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-udef 41.1%

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

   5. Applied egg-rr 41.1%

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

      1. expm1-def 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-log1p 44.9%

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

      3. associate-*r* 48.5%

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

   7. Simplified 48.5%

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

   if 0.0 < (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*))))) < +inf.0

   1. Initial program 69.4%

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

      1. associate-*l/ 74.6%

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

   3. Applied egg-rr 74.6%

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

   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 5.7%

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

   4. Taylor expanded in l around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. metadata-eval 27.8%

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

   6. Simplified 27.8%

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

   7. Taylor expanded in U* around inf 27.8%

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

      1. *-commutative 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in l around 0 17.3%

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

       1. sub-neg 17.3%

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

       2. associate-*r/ 17.3%

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

       3. metadata-eval 17.3%

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

       4. *-commutative 17.3%

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

       5. distribute-neg-frac 17.3%

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

       6. metadata-eval 17.3%

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

   12. Simplified 17.3%

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

4. Final simplification 62.2%

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

Alternative 3: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(l_m \cdot l_m\right)}{Om}\right) + t_1\right)\right)}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U*}{\frac{{Om}^{2}}{n}} + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(l_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_3 0.0)
     (sqrt (* (* 2.0 n) (* U (+ (- t (/ (* 2.0 (* l_m l_m)) Om)) t_1))))
     (if (<= t_3 2e+147)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (sqrt (* (+ (/ U* (/ (pow Om 2.0) n)) (/ -2.0 Om)) (* n U)))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + t_1))));
	} else if (t_3 <= 2e+147) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = sqrt((((U_42_ / (pow(Om, 2.0) / n)) + (-2.0 / Om)) * (n * U))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_2 = (2.0d0 * n) * u
    t_3 = sqrt((t_2 * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_1)))
    if (t_3 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t - ((2.0d0 * (l_m * l_m)) / om)) + t_1))))
    else if (t_3 <= 2d+147) then
        tmp = sqrt((t_2 * ((t - (2.0d0 * (l_m * (l_m / om)))) + t_1)))
    else
        tmp = sqrt((((u_42 / ((om ** 2.0d0) / n)) + ((-2.0d0) / om)) * (n * u))) * (l_m * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + t_1))));
	} else if (t_3 <= 2e+147) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = Math.sqrt((((U_42_ / (Math.pow(Om, 2.0) / n)) + (-2.0 / Om)) * (n * U))) * (l_m * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + t_1))))
	elif t_3 <= 2e+147:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = math.sqrt((((U_42_ / (math.pow(Om, 2.0) / n)) + (-2.0 / Om)) * (n * U))) * (l_m * math.sqrt(2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om)) + t_1))));
	elseif (t_3 <= 2e+147)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(U_42_ / Float64((Om ^ 2.0) / n)) + Float64(-2.0 / Om)) * Float64(n * U))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + t_1))));
	elseif (t_3 <= 2e+147)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = sqrt((((U_42_ / ((Om ^ 2.0) / n)) + (-2.0 / Om)) * (n * U))) * (l_m * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 2e+147], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(U$42$ / N[(N[Power[Om, 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * 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*))))) < 0.0

   1. Initial program 12.2%

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

   3. Simplified 44.9%

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

      1. expm1-log1p-u 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-udef 41.1%

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

   5. Applied egg-rr 41.1%

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

      1. expm1-def 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-log1p 44.9%

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

      3. associate-*r* 48.5%

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

   7. Simplified 48.5%

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

   if 0.0 < (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*))))) < 2e147

   1. Initial program 95.9%

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

      1. associate-*l/ 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 2e147 < (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 17.9%

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

   3. Simplified 26.4%

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

   4. Taylor expanded in l around inf 25.4%

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

      1. associate-*r/ 25.4%

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

      2. metadata-eval 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in U* around inf 25.6%

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

      1. *-commutative 25.6%

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

   9. Simplified 25.6%

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

   10. Taylor expanded in l around 0 21.0%

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

       1. associate-*r* 23.7%

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

       2. sub-neg 23.7%

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

       3. associate-*r/ 23.7%

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

       4. metadata-eval 23.7%

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

       5. associate-/l* 24.6%

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

       6. distribute-neg-frac 24.6%

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

       7. metadata-eval 24.6%

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

   12. Simplified 24.6%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U*}{\frac{{Om}^{2}}{n}} + \frac{-2}{Om}\right) \cdot \left(n \cdot U\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 4: 62.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.2%

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

   3. Simplified 44.9%

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

      1. expm1-log1p-u 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-udef 41.1%

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

   5. Applied egg-rr 41.1%

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

      1. expm1-def 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-log1p 44.9%

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

      3. associate-*r* 48.5%

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

   7. Simplified 48.5%

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

   if 0.0 < (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*))))) < +inf.0

   1. Initial program 69.4%

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

      1. associate-*l/ 74.6%

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

   3. Applied egg-rr 74.6%

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

   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 5.7%

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

   4. Taylor expanded in n around 0 4.0%

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

      1. add-sqr-sqrt 4.0%

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

      2. pow1/2 4.0%

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

      3. pow1/2 32.8%

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

      4. pow-prod-down 32.9%

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

   6. Applied egg-rr 32.9%

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

      1. unpow1/2 32.9%

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

      2. unpow2 32.9%

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

      3. rem-sqrt-square 32.9%

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

   8. Simplified 32.9%

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

   9. Taylor expanded in l around inf 40.3%

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

4. Final simplification 66.1%

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

Alternative 5: 62.4% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_3 0.0)
     (sqrt (* (* 2.0 n) (* U (+ (- t (/ (* 2.0 (* l_m l_m)) Om)) t_1))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (pow (* (* 2.0 U) (/ (* (pow l_m 2.0) -2.0) (/ Om n))) 0.5)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $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*))))) < 0.0

   1. Initial program 12.2%

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

   3. Simplified 44.9%

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

      1. expm1-log1p-u 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-udef 41.1%

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

   5. Applied egg-rr 41.1%

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

      1. expm1-def 41.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-log1p 44.9%

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

      3. associate-*r* 48.5%

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

   7. Simplified 48.5%

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

   if 0.0 < (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*))))) < +inf.0

   1. Initial program 69.4%

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

      1. associate-*l/ 74.6%

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

   3. Applied egg-rr 74.6%

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

   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 5.7%

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

   4. Taylor expanded in l around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. metadata-eval 27.8%

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

   6. Simplified 27.8%

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

   7. Taylor expanded in n around 0 13.5%

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

      1. associate-/l* 13.2%

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

   9. Simplified 13.2%

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

       1. pow1/2 39.7%

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

       2. associate-*r* 39.7%

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

       3. associate-*r/ 39.7%

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

   11. Applied egg-rr 39.7%

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

4. Final simplification 66.0%

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

Alternative 6: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(l_m \cdot l_m\right)}{Om}\right) + n \cdot \left({\left(\frac{l_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{if}\;l_m \leq 1.25 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;l_m \leq 1.9 \cdot 10^{-153}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;l_m \leq 1.85 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;l_m \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{\frac{U \cdot \left(n \cdot {l_m}^{2}\right)}{Om} \cdot -4 + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;l_m \leq 7.2 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 n)
           (*
            U
            (+
             (- t (/ (* 2.0 (* l_m l_m)) Om))
             (* n (* (pow (/ l_m Om) 2.0) (- U* U)))))))))
   (if (<= l_m 1.25e-211)
     t_1
     (if (<= l_m 1.9e-153)
       (pow (* (* (* 2.0 n) U) t) 0.5)
       (if (<= l_m 1.85e-82)
         t_1
         (if (<= l_m 1.1e-29)
           (sqrt
            (+
             (* (/ (* U (* n (pow l_m 2.0))) Om) -4.0)
             (* 2.0 (* U (* n t)))))
           (if (<= l_m 7.2e+146)
             t_1
             (* (* l_m (sqrt 2.0)) (sqrt (/ (* n (* U -2.0)) Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	double tmp;
	if (l_m <= 1.25e-211) {
		tmp = t_1;
	} else if (l_m <= 1.9e-153) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else if (l_m <= 1.85e-82) {
		tmp = t_1;
	} else if (l_m <= 1.1e-29) {
		tmp = sqrt(((((U * (n * pow(l_m, 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))));
	} else if (l_m <= 7.2e+146) {
		tmp = t_1;
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * (u * ((t - ((2.0d0 * (l_m * l_m)) / om)) + (n * (((l_m / om) ** 2.0d0) * (u_42 - u)))))))
    if (l_m <= 1.25d-211) then
        tmp = t_1
    else if (l_m <= 1.9d-153) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else if (l_m <= 1.85d-82) then
        tmp = t_1
    else if (l_m <= 1.1d-29) then
        tmp = sqrt(((((u * (n * (l_m ** 2.0d0))) / om) * (-4.0d0)) + (2.0d0 * (u * (n * t)))))
    else if (l_m <= 7.2d+146) then
        tmp = t_1
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * (u * (-2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (Math.pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	double tmp;
	if (l_m <= 1.25e-211) {
		tmp = t_1;
	} else if (l_m <= 1.9e-153) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else if (l_m <= 1.85e-82) {
		tmp = t_1;
	} else if (l_m <= 1.1e-29) {
		tmp = Math.sqrt(((((U * (n * Math.pow(l_m, 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))));
	} else if (l_m <= 7.2e+146) {
		tmp = t_1;
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (math.pow((l_m / Om), 2.0) * (U_42_ - U)))))))
	tmp = 0
	if l_m <= 1.25e-211:
		tmp = t_1
	elif l_m <= 1.9e-153:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	elif l_m <= 1.85e-82:
		tmp = t_1
	elif l_m <= 1.1e-29:
		tmp = math.sqrt(((((U * (n * math.pow(l_m, 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))))
	elif l_m <= 7.2e+146:
		tmp = t_1
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * (U * -2.0)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om)) + Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * Float64(U_42_ - U)))))))
	tmp = 0.0
	if (l_m <= 1.25e-211)
		tmp = t_1;
	elseif (l_m <= 1.9e-153)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	elseif (l_m <= 1.85e-82)
		tmp = t_1;
	elseif (l_m <= 1.1e-29)
		tmp = sqrt(Float64(Float64(Float64(Float64(U * Float64(n * (l_m ^ 2.0))) / Om) * -4.0) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l_m <= 7.2e+146)
		tmp = t_1;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * -2.0)) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (((l_m / Om) ^ 2.0) * (U_42_ - U)))))));
	tmp = 0.0;
	if (l_m <= 1.25e-211)
		tmp = t_1;
	elseif (l_m <= 1.9e-153)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	elseif (l_m <= 1.85e-82)
		tmp = t_1;
	elseif (l_m <= 1.1e-29)
		tmp = sqrt(((((U * (n * (l_m ^ 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))));
	elseif (l_m <= 7.2e+146)
		tmp = t_1;
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l$95$m, 1.25e-211], t$95$1, If[LessEqual[l$95$m, 1.9e-153], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 1.85e-82], t$95$1, If[LessEqual[l$95$m, 1.1e-29], N[Sqrt[N[(N[(N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 7.2e+146], t$95$1, N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.2500000000000001e-211 or 1.90000000000000011e-153 < l < 1.85e-82 or 1.09999999999999995e-29 < l < 7.1999999999999997e146

   1. Initial program 56.3%

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

   3. Simplified 55.1%

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

   if 1.2500000000000001e-211 < l < 1.90000000000000011e-153

   1. Initial program 49.4%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. associate-*r* 49.4%

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

      2. *-commutative 49.4%

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

   6. Simplified 49.4%

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

      1. pow1/2 49.4%

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

      2. associate-*r* 49.4%

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

      3. associate-*l* 49.4%

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

      4. *-commutative 49.4%

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

   8. Applied egg-rr 49.4%

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

   if 1.85e-82 < l < 1.09999999999999995e-29

   1. Initial program 63.3%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in Om around inf 63.2%

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

   if 7.1999999999999997e146 < l

   1. Initial program 12.7%

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

   3. Simplified 24.5%

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

   4. Taylor expanded in l around inf 52.3%

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

      1. associate-*r/ 52.3%

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

      2. metadata-eval 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in n around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. associate-*r* 45.2%

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

   9. Simplified 45.2%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-153}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{\frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} \cdot -4 + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \]

Alternative 7: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t - \frac{2 \cdot \left(l_m \cdot l_m\right)}{Om}\\ t_2 := {\left(\frac{l_m}{Om}\right)}^{2}\\ t_3 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t_1 + \left(n \cdot t_2\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{if}\;l_m \leq 2.05 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t_1 + n \cdot \left(t_2 \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;l_m \leq 2.3 \cdot 10^{-155}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;l_m \leq 1.16 \cdot 10^{-82}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;l_m \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\frac{U \cdot \left(n \cdot {l_m}^{2}\right)}{Om} \cdot -4 + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;l_m \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- t (/ (* 2.0 (* l_m l_m)) Om)))
        (t_2 (pow (/ l_m Om) 2.0))
        (t_3 (sqrt (* (* 2.0 n) (* U (+ t_1 (* (* n t_2) (- U* U))))))))
   (if (<= l_m 2.05e-209)
     (sqrt (* (* 2.0 n) (* U (+ t_1 (* n (* t_2 (- U* U)))))))
     (if (<= l_m 2.3e-155)
       (pow (* (* (* 2.0 n) U) t) 0.5)
       (if (<= l_m 1.16e-82)
         t_3
         (if (<= l_m 8e-30)
           (sqrt
            (+
             (* (/ (* U (* n (pow l_m 2.0))) Om) -4.0)
             (* 2.0 (* U (* n t)))))
           (if (<= l_m 5.2e+150)
             t_3
             (* (* l_m (sqrt 2.0)) (sqrt (/ (* n (* U -2.0)) Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t - ((2.0 * (l_m * l_m)) / Om);
	double t_2 = pow((l_m / Om), 2.0);
	double t_3 = sqrt(((2.0 * n) * (U * (t_1 + ((n * t_2) * (U_42_ - U))))));
	double tmp;
	if (l_m <= 2.05e-209) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else if (l_m <= 2.3e-155) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else if (l_m <= 1.16e-82) {
		tmp = t_3;
	} else if (l_m <= 8e-30) {
		tmp = sqrt(((((U * (n * pow(l_m, 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))));
	} else if (l_m <= 5.2e+150) {
		tmp = t_3;
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = t - ((2.0d0 * (l_m * l_m)) / om)
    t_2 = (l_m / om) ** 2.0d0
    t_3 = sqrt(((2.0d0 * n) * (u * (t_1 + ((n * t_2) * (u_42 - u))))))
    if (l_m <= 2.05d-209) then
        tmp = sqrt(((2.0d0 * n) * (u * (t_1 + (n * (t_2 * (u_42 - u)))))))
    else if (l_m <= 2.3d-155) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else if (l_m <= 1.16d-82) then
        tmp = t_3
    else if (l_m <= 8d-30) then
        tmp = sqrt(((((u * (n * (l_m ** 2.0d0))) / om) * (-4.0d0)) + (2.0d0 * (u * (n * t)))))
    else if (l_m <= 5.2d+150) then
        tmp = t_3
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * (u * (-2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t - ((2.0 * (l_m * l_m)) / Om);
	double t_2 = Math.pow((l_m / Om), 2.0);
	double t_3 = Math.sqrt(((2.0 * n) * (U * (t_1 + ((n * t_2) * (U_42_ - U))))));
	double tmp;
	if (l_m <= 2.05e-209) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else if (l_m <= 2.3e-155) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else if (l_m <= 1.16e-82) {
		tmp = t_3;
	} else if (l_m <= 8e-30) {
		tmp = Math.sqrt(((((U * (n * Math.pow(l_m, 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))));
	} else if (l_m <= 5.2e+150) {
		tmp = t_3;
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t - ((2.0 * (l_m * l_m)) / Om)
	t_2 = math.pow((l_m / Om), 2.0)
	t_3 = math.sqrt(((2.0 * n) * (U * (t_1 + ((n * t_2) * (U_42_ - U))))))
	tmp = 0
	if l_m <= 2.05e-209:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))))
	elif l_m <= 2.3e-155:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	elif l_m <= 1.16e-82:
		tmp = t_3
	elif l_m <= 8e-30:
		tmp = math.sqrt(((((U * (n * math.pow(l_m, 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))))
	elif l_m <= 5.2e+150:
		tmp = t_3
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * (U * -2.0)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om))
	t_2 = Float64(l_m / Om) ^ 2.0
	t_3 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + Float64(Float64(n * t_2) * Float64(U_42_ - U))))))
	tmp = 0.0
	if (l_m <= 2.05e-209)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + Float64(n * Float64(t_2 * Float64(U_42_ - U)))))));
	elseif (l_m <= 2.3e-155)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	elseif (l_m <= 1.16e-82)
		tmp = t_3;
	elseif (l_m <= 8e-30)
		tmp = sqrt(Float64(Float64(Float64(Float64(U * Float64(n * (l_m ^ 2.0))) / Om) * -4.0) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l_m <= 5.2e+150)
		tmp = t_3;
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * -2.0)) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = t - ((2.0 * (l_m * l_m)) / Om);
	t_2 = (l_m / Om) ^ 2.0;
	t_3 = sqrt(((2.0 * n) * (U * (t_1 + ((n * t_2) * (U_42_ - U))))));
	tmp = 0.0;
	if (l_m <= 2.05e-209)
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	elseif (l_m <= 2.3e-155)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	elseif (l_m <= 1.16e-82)
		tmp = t_3;
	elseif (l_m <= 8e-30)
		tmp = sqrt(((((U * (n * (l_m ^ 2.0))) / Om) * -4.0) + (2.0 * (U * (n * t)))));
	elseif (l_m <= 5.2e+150)
		tmp = t_3;
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l$95$m, 2.05e-209], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.3e-155], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 1.16e-82], t$95$3, If[LessEqual[l$95$m, 8e-30], N[Sqrt[N[(N[(N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 5.2e+150], t$95$3, N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 2.04999999999999989e-209

   1. Initial program 55.7%

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

   3. Simplified 53.6%

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

   if 2.04999999999999989e-209 < l < 2.30000000000000005e-155

   1. Initial program 49.4%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in l around 0 44.4%

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

      1. associate-*r* 49.4%

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

      2. *-commutative 49.4%

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

   6. Simplified 49.4%

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

      1. pow1/2 49.4%

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

      2. associate-*r* 49.4%

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

      3. associate-*l* 49.4%

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

      4. *-commutative 49.4%

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

   8. Applied egg-rr 49.4%

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

   if 2.30000000000000005e-155 < l < 1.16e-82 or 8.000000000000001e-30 < l < 5.20000000000000012e150

   1. Initial program 58.4%

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

   3. Simplified 60.5%

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

      1. expm1-log1p-u 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-udef 44.6%

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

   5. Applied egg-rr 44.6%

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

      1. expm1-def 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      2. expm1-log1p 60.5%

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

      3. associate-*r* 62.6%

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

   7. Simplified 62.6%

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

   if 1.16e-82 < l < 8.000000000000001e-30

   1. Initial program 63.3%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in Om around inf 63.2%

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

   if 5.20000000000000012e150 < l

   1. Initial program 12.7%

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

   3. Simplified 24.5%

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

   4. Taylor expanded in l around inf 52.3%

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

      1. associate-*r/ 52.3%

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

      2. metadata-eval 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in n around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. associate-*r* 45.2%

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

   9. Simplified 45.2%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-155}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.16 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} \cdot -4 + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \]

Alternative 8: 48.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 3 \cdot 10^{-137}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;l_m \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{l_m}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3e-137)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (if (<= l_m 5.2e+150)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l_m 2.0) Om)))))))
     (* (* l_m (sqrt 2.0)) (sqrt (/ (* n (* U -2.0)) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3e-137) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else if (l_m <= 5.2e+150) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 3d-137) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else if (l_m <= 5.2d+150) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * (u * (-2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3e-137) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else if (l_m <= 5.2e+150) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 3e-137:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	elif l_m <= 5.2e+150:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * (U * -2.0)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3e-137)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	elseif (l_m <= 5.2e+150)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * -2.0)) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 3e-137)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	elseif (l_m <= 5.2e+150)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l_m ^ 2.0) / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3e-137], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 5.2e+150], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.9999999999999998e-137

   1. Initial program 55.2%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in l around 0 44.8%

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

      1. associate-*r* 47.3%

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

      2. *-commutative 47.3%

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

   6. Simplified 47.3%

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

      1. pow1/2 49.0%

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

      2. associate-*r* 49.1%

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

      3. associate-*l* 49.1%

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

      4. *-commutative 49.1%

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

   8. Applied egg-rr 49.1%

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

   if 2.9999999999999998e-137 < l < 5.20000000000000012e150

   1. Initial program 59.7%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in n around 0 52.0%

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

   if 5.20000000000000012e150 < l

   1. Initial program 12.7%

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

   3. Simplified 24.5%

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

   4. Taylor expanded in l around inf 52.3%

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

      1. associate-*r/ 52.3%

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

      2. metadata-eval 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in n around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. associate-*r* 45.2%

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

   9. Simplified 45.2%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-137}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \]

Alternative 9: 48.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 6.2 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(l_m \cdot \frac{l_m}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.2e+181)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l_m (/ l_m Om)) -2.0)))))
   (* (* l_m (sqrt 2.0)) (sqrt (/ (* n (* U -2.0)) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 6.2e+181) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 6.2d+181) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m * (l_m / om)) * (-2.0d0))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * (u * (-2.0d0))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 6.2e+181) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * (U * -2.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 6.2e+181:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * (U * -2.0)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.2e+181)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m * Float64(l_m / Om)) * -2.0)))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * -2.0)) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 6.2e+181)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m * (l_m / Om)) * -2.0)))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * (U * -2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 6.2e+181], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.19999999999999978e181

   1. Initial program 54.7%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in n around 0 48.1%

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

      1. unpow2 48.1%

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

      2. associate-*l/ 51.8%

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

   6. Applied egg-rr 51.8%

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

   if 6.19999999999999978e181 < l

   1. Initial program 16.5%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in l around inf 55.3%

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

      1. associate-*r/ 55.3%

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

      2. metadata-eval 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in n around 0 50.6%

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

      1. associate-*r/ 50.6%

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

      2. associate-*r* 50.6%

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

   9. Simplified 50.6%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot -2\right)}{Om}}\\ \end{array} \]

Alternative 10: 46.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Simplified 54.8%

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

4. Taylor expanded in n around 0 45.7%

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

   1. unpow2 45.7%

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

   2. associate-*l/ 49.6%

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

6. Applied egg-rr 49.6%

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

7. Final simplification 49.6%

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

Alternative 11: 37.9% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* (* (* 2.0 n) U) t) 0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Simplified 54.8%

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

4. Taylor expanded in l around 0 39.0%

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

   1. associate-*r* 40.2%

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

   2. *-commutative 40.2%

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

6. Simplified 40.2%

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

   1. pow1/2 43.0%

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

   2. associate-*r* 43.0%

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

   3. associate-*l* 43.0%

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

   4. *-commutative 43.0%

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

8. Applied egg-rr 43.0%

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

9. Final simplification 43.0%

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

Alternative 12: 4.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Simplified 54.8%

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

4. Taylor expanded in l around 0 39.0%

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

   1. add-sqr-sqrt 39.0%

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

   2. pow1/2 39.0%

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

   3. pow1/2 40.9%

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

   4. pow-prod-down 31.1%

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

   5. pow2 31.1%

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

   6. associate-*r* 31.8%

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

   7. *-commutative 31.8%

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

   8. associate-*l* 30.3%

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

6. Applied egg-rr 30.3%

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

7. Taylor expanded in n around -inf 4.4%

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

8. Final simplification 4.4%

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

Alternative 13: 36.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Simplified 54.8%

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

4. Taylor expanded in l around 0 39.0%

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

5. Final simplification 39.0%

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

Alternative 14: 36.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Simplified 54.8%

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

4. Taylor expanded in l around 0 39.0%

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

   1. associate-*r* 40.2%

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

   2. *-commutative 40.2%

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

6. Simplified 40.2%

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

7. Final simplification 40.2%

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

Reproduce

herbie shell --seed 2023322 
(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*))))))