Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 64.5%

Time: 25.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% 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: 64.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(U* - U, \frac{n}{{Om}^{2}}, \frac{-2}{Om}\right)\right)}\\ \end{array} \end{array} \]

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)
     (exp (* (- (log (* -2.0 (* n t))) (log (/ -1.0 U))) 0.5))
     (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 (fma (- U* U) (/ n (pow Om 2.0)) (/ -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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	} 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 * fma((U_42_ - U), (n / pow(Om, 2.0)), (-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 = exp(Float64(Float64(log(Float64(-2.0 * Float64(n * t))) - log(Float64(-1.0 / U))) * 0.5));
	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 * fma(Float64(U_42_ - U), Float64(n / (Om ^ 2.0)), Float64(-2.0 / Om))))));
	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[(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[Exp[N[(N[(N[Log[N[(-2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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[(U$42$ - U), $MachinePrecision] * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $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 5.1%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around inf 39.5%

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

      1. pow1/2 39.5%

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

      2. pow-to-exp 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around -inf 49.0%

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

      1. mul-1-neg 49.0%

         \[\leadsto e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{U}\right)\right)}\right) \cdot 0.5} \]

      2. unsub-neg 49.0%

         \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   9. Simplified 49.0%

      \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   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 67.9%

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

      1. associate-*l/ 74.3%

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

   3. Applied egg-rr 74.3%

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

   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 4.9%

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

   4. Taylor expanded in l around inf 28.7%

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

      1. associate-/l* 26.7%

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

      2. associate-*r/ 26.7%

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

      3. metadata-eval 26.7%

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

   6. Simplified 26.7%

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

   7. Taylor expanded in l around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. associate-*l/ 34.2%

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

      3. *-commutative 34.2%

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

      4. fma-neg 34.2%

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

      5. associate-*r/ 34.2%

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

      6. metadata-eval 34.2%

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

      7. distribute-neg-frac 34.2%

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

      8. metadata-eval 34.2%

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

   9. Simplified 34.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(U* - U, \frac{n}{{Om}^{2}}, \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 2: 64.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \end{array} \]

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)
     (exp (* (- (log (* -2.0 (* n t))) (log (/ -1.0 U))) 0.5))
     (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 (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	} 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 * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / 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.exp(((Math.log((-2.0 * (n * t))) - Math.log((-1.0 / U))) * 0.5));
	} 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 * (((n * (U_42_ - U)) / Math.pow(Om, 2.0)) + (2.0 * (-1.0 / 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.exp(((math.log((-2.0 * (n * t))) - math.log((-1.0 / U))) * 0.5))
	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 * (((n * (U_42_ - U)) / math.pow(Om, 2.0)) + (2.0 * (-1.0 / 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 = exp(Float64(Float64(log(Float64(-2.0 * Float64(n * t))) - log(Float64(-1.0 / U))) * 0.5));
	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(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / 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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	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 * (((n * (U_42_ - U)) / (Om ^ 2.0)) + (2.0 * (-1.0 / 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[Exp[N[(N[(N[Log[N[(-2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 5.1%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around inf 39.5%

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

      1. pow1/2 39.5%

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

      2. pow-to-exp 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around -inf 49.0%

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

      1. mul-1-neg 49.0%

         \[\leadsto e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{U}\right)\right)}\right) \cdot 0.5} \]

      2. unsub-neg 49.0%

         \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   9. Simplified 49.0%

      \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   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 67.9%

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

      1. associate-*l/ 74.3%

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

   3. Applied egg-rr 74.3%

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

   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 4.9%

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

   4. Taylor expanded in l around inf 34.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \]

Alternative 3: 63.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

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)
     (exp (* (- (log (* -2.0 (* n t))) (log (/ -1.0 U))) 0.5))
     (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 (- (/ n (/ (pow Om 2.0) (- U* 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 = (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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	} 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 * ((n / (pow(Om, 2.0) / (U_42_ - U))) - (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.exp(((Math.log((-2.0 * (n * t))) - Math.log((-1.0 / U))) * 0.5));
	} 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 * ((n / (Math.pow(Om, 2.0) / (U_42_ - U))) - (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.exp(((math.log((-2.0 * (n * t))) - math.log((-1.0 / U))) * 0.5))
	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 * ((n / (math.pow(Om, 2.0) / (U_42_ - U))) - (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 = exp(Float64(Float64(log(Float64(-2.0 * Float64(n * t))) - log(Float64(-1.0 / U))) * 0.5));
	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(n / Float64((Om ^ 2.0) / Float64(U_42_ - U))) - Float64(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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	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 * ((n / ((Om ^ 2.0) / (U_42_ - 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[(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[Exp[N[(N[(N[Log[N[(-2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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[(n / N[(N[Power[Om, 2.0], $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (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 5.1%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around inf 39.5%

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

      1. pow1/2 39.5%

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

      2. pow-to-exp 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around -inf 49.0%

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

      1. mul-1-neg 49.0%

         \[\leadsto e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{U}\right)\right)}\right) \cdot 0.5} \]

      2. unsub-neg 49.0%

         \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   9. Simplified 49.0%

      \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   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 67.9%

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

      1. associate-*l/ 74.3%

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

   3. Applied egg-rr 74.3%

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

   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 4.9%

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

   4. Taylor expanded in l around inf 34.2%

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

      1. associate-/l* 32.1%

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

      2. associate-*r/ 32.1%

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

      3. metadata-eval 32.1%

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

   6. Simplified 32.1%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{n}{\frac{{Om}^{2}}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 4: 58.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(-U\right)}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

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)
     (exp (* (- (log (* -2.0 (* n t))) (log (/ -1.0 U))) 0.5))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (- (/ (* n (- U)) (pow Om 2.0)) (/ 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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	} 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(((n * U) * (((n * -U) / pow(Om, 2.0)) - (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.exp(((Math.log((-2.0 * (n * t))) - Math.log((-1.0 / U))) * 0.5));
	} 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(((n * U) * (((n * -U) / Math.pow(Om, 2.0)) - (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.exp(((math.log((-2.0 * (n * t))) - math.log((-1.0 / U))) * 0.5))
	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(((n * U) * (((n * -U) / math.pow(Om, 2.0)) - (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 = exp(Float64(Float64(log(Float64(-2.0 * Float64(n * t))) - log(Float64(-1.0 / U))) * 0.5));
	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(Float64(n * U) * Float64(Float64(Float64(n * Float64(-U)) / (Om ^ 2.0)) - Float64(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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	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(((n * U) * (((n * -U) / (Om ^ 2.0)) - (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[Exp[N[(N[(N[Log[N[(-2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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[(N[(n * U), $MachinePrecision] * N[(N[(N[(n * (-U)), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / 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 5.1%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around inf 39.5%

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

      1. pow1/2 39.5%

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

      2. pow-to-exp 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around -inf 49.0%

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

      1. mul-1-neg 49.0%

         \[\leadsto e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{U}\right)\right)}\right) \cdot 0.5} \]

      2. unsub-neg 49.0%

         \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   9. Simplified 49.0%

      \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   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 67.9%

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

      1. associate-*l/ 74.3%

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

   3. Applied egg-rr 74.3%

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

   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 4.9%

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

   4. Taylor expanded in l around inf 28.7%

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

      1. associate-/l* 26.7%

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

      2. associate-*r/ 26.7%

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

      3. metadata-eval 26.7%

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

   6. Simplified 26.7%

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

   7. Taylor expanded in U* around 0 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-*r* 19.6%

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

      3. mul-1-neg 19.6%

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

      4. associate-*r/ 19.6%

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

      5. metadata-eval 19.6%

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

   9. Simplified 19.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(-U\right)}{{Om}^{2}} - \frac{2}{Om}\right)}\\ \end{array} \]

Alternative 5: 60.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

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)
     (exp (* (- (log (* -2.0 (* n t))) (log (/ -1.0 U))) 0.5))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l_m 2.0)) Om)))) 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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	} 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 * ((n * U) * (t - ((2.0 * pow(l_m, 2.0)) / Om)))), 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.exp(((Math.log((-2.0 * (n * t))) - Math.log((-1.0 / U))) * 0.5));
	} 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 * ((n * U) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))), 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.exp(((math.log((-2.0 * (n * t))) - math.log((-1.0 / U))) * 0.5))
	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 * ((n * U) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))), 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 = exp(Float64(Float64(log(Float64(-2.0 * Float64(n * t))) - log(Float64(-1.0 / U))) * 0.5));
	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(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))) ^ 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 = exp(((log((-2.0 * (n * t))) - log((-1.0 / U))) * 0.5));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l_m ^ 2.0)) / Om)))) ^ 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[Exp[N[(N[(N[Log[N[(-2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $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 5.1%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around inf 39.5%

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

      1. pow1/2 39.5%

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

      2. pow-to-exp 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around -inf 49.0%

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

      1. mul-1-neg 49.0%

         \[\leadsto e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{U}\right)\right)}\right) \cdot 0.5} \]

      2. unsub-neg 49.0%

         \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   9. Simplified 49.0%

      \[\leadsto e^{\color{blue}{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right)} \cdot 0.5} \]

   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 67.9%

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

      1. associate-*l/ 74.3%

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

   3. Applied egg-rr 74.3%

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

   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%

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

   2. Taylor expanded in n around 0 8.8%

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

      1. pow1/2 30.3%

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

      2. associate-*r* 29.4%

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

      3. *-commutative 29.4%

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

      4. associate-*r/ 29.4%

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

   4. Applied egg-rr 29.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;e^{\left(\log \left(-2 \cdot \left(n \cdot t\right)\right) - \log \left(\frac{-1}{U}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 6: 62.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t - 2 \cdot \left(l_m \cdot \frac{l_m}{Om}\right)\\ t_2 := \left(n \cdot {\left(\frac{l_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t - 2 \cdot \frac{l_m \cdot l_m}{Om}\right) + t_2\right)\\ \mathbf{if}\;t_4 \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t_1\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(t_1 + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \left(n \cdot \frac{{l_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (* l_m (/ l_m Om)))))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2))))
   (if (<= t_4 5e-278)
     (sqrt (* 2.0 (* U (* n t_1))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ t_1 t_2)))
       (pow (* 2.0 (* (* U -2.0) (* n (/ (pow l_m 2.0) Om)))) 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 = t - (2.0 * (l_m * (l_m / Om)));
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double tmp;
	if (t_4 <= 5e-278) {
		tmp = sqrt((2.0 * (U * (n * t_1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (t_1 + t_2)));
	} else {
		tmp = pow((2.0 * ((U * -2.0) * (n * (pow(l_m, 2.0) / Om)))), 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 = t - (2.0 * (l_m * (l_m / Om)));
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double tmp;
	if (t_4 <= 5e-278) {
		tmp = Math.sqrt((2.0 * (U * (n * t_1))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * (t_1 + t_2)));
	} else {
		tmp = Math.pow((2.0 * ((U * -2.0) * (n * (Math.pow(l_m, 2.0) / Om)))), 0.5);
	}
	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 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)
	tmp = 0
	if t_4 <= 5e-278:
		tmp = math.sqrt((2.0 * (U * (n * t_1))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * (t_1 + t_2)))
	else:
		tmp = math.pow((2.0 * ((U * -2.0) * (n * (math.pow(l_m, 2.0) / Om)))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om))))
	t_2 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2))
	tmp = 0.0
	if (t_4 <= 5e-278)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_1))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(t_1 + t_2)));
	else
		tmp = Float64(2.0 * Float64(Float64(U * -2.0) * Float64(n * Float64((l_m ^ 2.0) / Om)))) ^ 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 = t - (2.0 * (l_m * (l_m / Om)));
	t_2 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	tmp = 0.0;
	if (t_4 <= 5e-278)
		tmp = sqrt((2.0 * (U * (n * t_1))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * (t_1 + t_2)));
	else
		tmp = (2.0 * ((U * -2.0) * (n * ((l_m ^ 2.0) / Om)))) ^ 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[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-278], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(U * -2.0), $MachinePrecision] * N[(n * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.99999999999999985e-278

   1. Initial program 9.4%

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

   2. Taylor expanded in n around 0 39.5%

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

      1. unpow2 39.5%

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

      2. associate-*l/ 42.3%

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

   4. Applied egg-rr 42.3%

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

   if 4.99999999999999985e-278 < (*.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 68.6%

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

      1. associate-*l/ 75.1%

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

   3. Applied egg-rr 75.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around 0 5.3%

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

   3. Taylor expanded in t around 0 2.8%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1/2 27.5%

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

      2. associate-*r* 27.5%

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

      3. associate-/r/ 27.8%

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

   7. Applied egg-rr 27.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \left(n \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l_m \cdot \frac{l_m}{Om}\\ \mathbf{if}\;n \leq -2.3 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(l_m \cdot l_m\right)}{Om}\right) + n \cdot \left({\left(\frac{l_m}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot t_1\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om))))
   (if (<= n -2.3e+103)
     (sqrt
      (*
       (* 2.0 n)
       (*
        U
        (+
         (- t (/ (* 2.0 (* l_m l_m)) Om))
         (* n (* (pow (/ l_m Om) 2.0) (- U* U)))))))
     (if (<= n 1.1e-209)
       (sqrt (* 2.0 (* U (* n (- t (* 2.0 t_1))))))
       (if (<= n 3.8e+49)
         (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 t_1)))))
         (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l_m 2.0)) Om)))) 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 = l_m * (l_m / Om);
	double tmp;
	if (n <= -2.3e+103) {
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (pow((l_m / Om), 2.0) * (U_42_ - U)))))));
	} else if (n <= 1.1e-209) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else if (n <= 3.8e+49) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	} else {
		tmp = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l_m, 2.0)) / Om)))), 0.5);
	}
	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 = l_m * (l_m / om)
    if (n <= (-2.3d+103)) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t - ((2.0d0 * (l_m * l_m)) / om)) + (n * (((l_m / om) ** 2.0d0) * (u_42 - u)))))))
    else if (n <= 1.1d-209) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * t_1))))))
    else if (n <= 3.8d+49) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * t_1))))
    else
        tmp = (2.0d0 * ((n * u) * (t - ((2.0d0 * (l_m ** 2.0d0)) / om)))) ** 0.5d0
    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 = l_m * (l_m / Om);
	double tmp;
	if (n <= -2.3e+103) {
		tmp = 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)))))));
	} else if (n <= 1.1e-209) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else if (n <= 3.8e+49) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * t_1))));
	} else {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	tmp = 0
	if n <= -2.3e+103:
		tmp = 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)))))))
	elif n <= 1.1e-209:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))))
	elif n <= 3.8e+49:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * t_1))))
	else:
		tmp = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m * Float64(l_m / Om))
	tmp = 0.0
	if (n <= -2.3e+103)
		tmp = 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)))))));
	elseif (n <= 1.1e-209)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * t_1))))));
	elseif (n <= 3.8e+49)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * t_1)))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))) ^ 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 = l_m * (l_m / Om);
	tmp = 0.0;
	if (n <= -2.3e+103)
		tmp = sqrt(((2.0 * n) * (U * ((t - ((2.0 * (l_m * l_m)) / Om)) + (n * (((l_m / Om) ^ 2.0) * (U_42_ - U)))))));
	elseif (n <= 1.1e-209)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	elseif (n <= 3.8e+49)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	else
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l_m ^ 2.0)) / Om)))) ^ 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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.3e+103], 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[n, 1.1e-209], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.8e+49], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.30000000000000008e103

   1. Initial program 47.3%

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

   3. Simplified 44.4%

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

   if -2.30000000000000008e103 < n < 1.10000000000000005e-209

   1. Initial program 50.6%

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

   2. Taylor expanded in n around 0 53.6%

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

      1. unpow2 53.6%

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

      2. associate-*l/ 61.4%

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

   4. Applied egg-rr 61.4%

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

   if 1.10000000000000005e-209 < n < 3.7999999999999999e49

   1. Initial program 42.3%

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

   3. Simplified 60.7%

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

      1. sqrt-prod 67.1%

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

      2. fma-udef 67.1%

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

      3. associate-*l/ 56.4%

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

      4. associate-*r/ 56.4%

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

      5. associate--l- 56.4%

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

      6. associate--l- 56.4%

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

   5. Applied egg-rr 56.4%

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

      1. *-commutative 56.4%

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

      2. rem-3cbrt-rft 56.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \color{blue}{\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}\right)} \]

      3. unpow2 56.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}^{2}}\right)} \]

      4. cancel-sign-sub-inv 56.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \color{blue}{\left(t + \left(-\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right) \cdot {\left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}^{2}\right)}} \]

   7. Simplified 56.5%

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

   8. Taylor expanded in n around 0 55.1%

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

      1. unpow2 45.1%

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

      2. associate-*l/ 54.0%

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

   10. Applied egg-rr 65.6%

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

   if 3.7999999999999999e49 < n

   1. Initial program 52.3%

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

   2. Taylor expanded in n around 0 35.9%

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

      1. pow1/2 49.8%

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

      2. associate-*r* 57.8%

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

      3. *-commutative 57.8%

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

      4. associate-*r/ 57.8%

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

   4. Applied egg-rr 57.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 8: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l_m \cdot \frac{l_m}{Om}\\ t_2 := {\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot t_1\right)\right)\right)}\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om)))
        (t_2 (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l_m 2.0)) Om)))) 0.5)))
   (if (<= n -1.35e+68)
     t_2
     (if (<= n 2.3e-209)
       (sqrt (* 2.0 (* U (* n (- t (* 2.0 t_1))))))
       (if (<= n 8.6e+49)
         (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 t_1)))))
         t_2)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double t_2 = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l_m, 2.0)) / Om)))), 0.5);
	double tmp;
	if (n <= -1.35e+68) {
		tmp = t_2;
	} else if (n <= 2.3e-209) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else if (n <= 8.6e+49) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = l_m * (l_m / om)
    t_2 = (2.0d0 * ((n * u) * (t - ((2.0d0 * (l_m ** 2.0d0)) / om)))) ** 0.5d0
    if (n <= (-1.35d+68)) then
        tmp = t_2
    else if (n <= 2.3d-209) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * t_1))))))
    else if (n <= 8.6d+49) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * t_1))))
    else
        tmp = t_2
    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 = l_m * (l_m / Om);
	double t_2 = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))), 0.5);
	double tmp;
	if (n <= -1.35e+68) {
		tmp = t_2;
	} else if (n <= 2.3e-209) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else if (n <= 8.6e+49) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	t_2 = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))), 0.5)
	tmp = 0
	if n <= -1.35e+68:
		tmp = t_2
	elif n <= 2.3e-209:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))))
	elif n <= 8.6e+49:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * t_1))))
	else:
		tmp = t_2
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m * Float64(l_m / Om))
	t_2 = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5
	tmp = 0.0
	if (n <= -1.35e+68)
		tmp = t_2;
	elseif (n <= 2.3e-209)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * t_1))))));
	elseif (n <= 8.6e+49)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = l_m * (l_m / Om);
	t_2 = (2.0 * ((n * U) * (t - ((2.0 * (l_m ^ 2.0)) / Om)))) ^ 0.5;
	tmp = 0.0;
	if (n <= -1.35e+68)
		tmp = t_2;
	elseif (n <= 2.3e-209)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	elseif (n <= 8.6e+49)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	else
		tmp = t_2;
	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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[n, -1.35e+68], t$95$2, If[LessEqual[n, 2.3e-209], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 8.6e+49], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999995e68 or 8.5999999999999998e49 < n

   1. Initial program 51.0%

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

   2. Taylor expanded in n around 0 35.6%

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

      1. pow1/2 48.4%

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

      2. associate-*r* 53.3%

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

      3. *-commutative 53.3%

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

      4. associate-*r/ 53.3%

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

   4. Applied egg-rr 53.3%

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

   if -1.34999999999999995e68 < n < 2.3e-209

   1. Initial program 50.2%

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

   2. Taylor expanded in n around 0 53.5%

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

      1. unpow2 53.5%

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

      2. associate-*l/ 61.8%

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

   4. Applied egg-rr 61.8%

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

   if 2.3e-209 < n < 8.5999999999999998e49

   1. Initial program 42.3%

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

   3. Simplified 60.7%

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

      1. sqrt-prod 67.1%

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

      2. fma-udef 67.1%

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

      3. associate-*l/ 56.4%

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

      4. associate-*r/ 56.4%

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

      5. associate--l- 56.4%

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

      6. associate--l- 56.4%

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

   5. Applied egg-rr 56.4%

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

      1. *-commutative 56.4%

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

      2. rem-3cbrt-rft 56.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \color{blue}{\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}\right)} \]

      3. unpow2 56.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}^{2}}\right)} \]

      4. cancel-sign-sub-inv 56.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \color{blue}{\left(t + \left(-\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right) \cdot {\left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}^{2}\right)}} \]

   7. Simplified 56.5%

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

   8. Taylor expanded in n around 0 55.1%

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

      1. unpow2 45.1%

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

      2. associate-*l/ 54.0%

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

   10. Applied egg-rr 65.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{+68}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 9: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l_m \cdot \frac{l_m}{Om}\\ \mathbf{if}\;n \leq 2 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om))))
   (if (<= n 2e-209)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 t_1))))))
     (* (sqrt (* 2.0 n)) (sqrt (* U (+ t (* -2.0 t_1))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double tmp;
	if (n <= 2e-209) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	}
	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 = l_m * (l_m / om)
    if (n <= 2d-209) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * t_1))))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t + ((-2.0d0) * t_1))))
    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 = l_m * (l_m / Om);
	double tmp;
	if (n <= 2e-209) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t + (-2.0 * t_1))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = l_m * (l_m / Om)
	tmp = 0
	if n <= 2e-209:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t + (-2.0 * t_1))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m * Float64(l_m / Om))
	tmp = 0.0
	if (n <= 2e-209)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * t_1))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * t_1)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = l_m * (l_m / Om);
	tmp = 0.0;
	if (n <= 2e-209)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + (-2.0 * t_1))));
	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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 2e-209], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 2.0000000000000001e-209

   1. Initial program 49.9%

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

   2. Taylor expanded in n around 0 48.7%

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

      1. unpow2 48.7%

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

      2. associate-*l/ 54.8%

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

   4. Applied egg-rr 54.8%

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

   if 2.0000000000000001e-209 < n

   1. Initial program 47.5%

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

   3. Simplified 58.0%

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

      1. sqrt-prod 64.4%

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

      2. fma-udef 64.4%

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

      3. associate-*l/ 57.6%

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

      4. associate-*r/ 57.6%

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

      5. associate--l- 57.6%

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

      6. associate--l- 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. *-commutative 57.6%

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

      2. rem-3cbrt-rft 57.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \color{blue}{\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}\right)} \]

      3. unpow2 57.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}^{2}}\right)} \]

      4. cancel-sign-sub-inv 57.5%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \color{blue}{\left(t + \left(-\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right) \cdot {\left(\sqrt[3]{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)}^{2}\right)}} \]

   7. Simplified 56.8%

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

   8. Taylor expanded in n around 0 46.0%

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

      1. unpow2 40.4%

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

      2. associate-*l/ 44.6%

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

   10. Applied egg-rr 52.8%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 10: 48.9% accurate, 2.0× speedup

Math

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

FPCore

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

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 - (2.0 * (l_m * (l_m / Om))))))));
}

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 - (2.0d0 * (l_m * (l_m / om))))))))
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 - (2.0 * (l_m * (l_m / Om))))))));
}

Python

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

Julia

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

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l_m * (l_m / Om))))))));
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 * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.8%

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

2. Taylor expanded in n around 0 45.0%

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

   1. unpow2 45.0%

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

   2. associate-*l/ 50.3%

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

4. Applied egg-rr 50.3%

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

5. Final simplification 50.3%

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

Alternative 11: 35.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l_m \leq 2.9 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.9e-227)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* (* 2.0 n) (* U t)))))

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 <= 2.9e-227) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * t)));
	}
	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 <= 2.9d-227) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    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 <= 2.9e-227) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.9e-227:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.9e-227)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	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 <= 2.9e-227)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt(((2.0 * n) * (U * t)));
	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, 2.9e-227], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.90000000000000011e-227

   1. Initial program 50.2%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in l around 0 35.7%

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

   if 2.90000000000000011e-227 < l

   1. Initial program 47.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in t around inf 33.5%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 12: 38.3% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* (* n t) (* 2.0 U)) 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(((n * t) * (2.0 * U)), 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 = ((n * t) * (2.0d0 * u)) ** 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(((n * t) * (2.0 * U)), 0.5);
}

Python

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

Julia

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

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = ((n * t) * (2.0 * U)) ^ 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 * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 48.8%

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

3. Simplified 54.1%

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

4. Taylor expanded in l around 0 34.5%

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

   1. pow1/2 37.4%

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

   2. associate-*r* 37.4%

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

6. Applied egg-rr 37.4%

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

7. Final simplification 37.4%

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

Alternative 13: 36.4% 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 48.8%

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

3. Simplified 54.1%

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

4. Taylor expanded in l around 0 34.5%

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

5. Final simplification 34.5%

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

Reproduce

herbie shell --seed 2023339 
(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*))))))