Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 63.0%

Time: 24.0s

Alternatives: 16

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_4 0.0)
     (sqrt (* 2.0 (* U (* n (- (fma t_1 -2.0 t) (* t_1 (/ (* n U) Om)))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (sqrt
        (*
         (* n -2.0)
         (* (* U (* l l)) (- (/ 2.0 Om) (* (/ n Om) (/ U* Om))))))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.2%

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

      1. associate-*l/ 8.5%

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

   3. Applied egg-rr 8.5%

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

   4. Taylor expanded in U* around 0 30.3%

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

      1. associate-*r* 30.3%

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

   6. Simplified 37.2%

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

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

   1. Initial program 72.4%

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

      1. associate-*l/ 74.4%

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

   3. Applied egg-rr 74.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.3%

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

   4. Taylor expanded in l around inf 50.8%

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

      1. unpow2 50.8%

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

      2. associate-*r/ 50.8%

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

      3. metadata-eval 50.8%

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

      4. *-commutative 50.8%

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

      5. unpow2 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in U around 0 50.5%

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

      1. unpow2 50.5%

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

      2. *-commutative 50.5%

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

      3. fma-def 50.5%

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

      4. unpow2 50.5%

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

      5. associate-*r/ 50.5%

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

      6. metadata-eval 50.5%

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

   9. Simplified 50.5%

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

       1. *-un-lft-identity 50.5%

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

       2. times-frac 59.2%

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

   11. Applied egg-rr 59.2%

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

       1. *-lft-identity 59.2%

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

       2. associate-*r* 59.2%

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

       3. unpow2 59.2%

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

       4. associate-*r* 59.5%

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

       5. *-commutative 59.5%

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

       6. unpow2 59.5%

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

       7. times-frac 50.8%

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

       8. unpow2 50.8%

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

       9. fma-def 50.8%

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

       10. +-commutative 50.8%

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

       11. mul-1-neg 50.8%

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

       12. unpow2 50.8%

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

       13. times-frac 59.5%

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

       14. unsub-neg 59.5%

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

   13. Simplified 59.5%

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

4. Final simplification 67.9%

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

Alternative 2: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ t_3 := \frac{U* - U}{Om}\\ \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left(t_1 \cdot t_3\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om} + \mathsf{fma}\left(-2, t_1, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot t_3 - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))) (t_2 (* 2.0 (* n U))) (t_3 (/ (- U* U) Om)))
   (if (<= l 9.5e-28)
     (sqrt (* t_2 (+ (+ t (* (/ (* l l) Om) -2.0)) (* n (* t_1 t_3)))))
     (if (<= l 7.4e+18)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ (* (/ n Om) (/ (* (* l l) U*) Om)) (fma -2.0 t_1 t)))))
       (sqrt (* t_2 (+ t (* (* l l) (- (* (/ n Om) t_3) (/ 2.0 Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = 2.0 * (n * U);
	double t_3 = (U_42_ - U) / Om;
	double tmp;
	if (l <= 9.5e-28) {
		tmp = sqrt((t_2 * ((t + (((l * l) / Om) * -2.0)) + (n * (t_1 * t_3)))));
	} else if (l <= 7.4e+18) {
		tmp = sqrt(((2.0 * n) * (U * (((n / Om) * (((l * l) * U_42_) / Om)) + fma(-2.0, t_1, t)))));
	} else {
		tmp = sqrt((t_2 * (t + ((l * l) * (((n / Om) * t_3) - (2.0 / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(2.0 * Float64(n * U))
	t_3 = Float64(Float64(U_42_ - U) / Om)
	tmp = 0.0
	if (l <= 9.5e-28)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(n * Float64(t_1 * t_3)))));
	elseif (l <= 7.4e+18)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(Float64(n / Om) * Float64(Float64(Float64(l * l) * U_42_) / Om)) + fma(-2.0, t_1, t)))));
	else
		tmp = sqrt(Float64(t_2 * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n / Om) * t_3) - Float64(2.0 / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, 9.5e-28], N[Sqrt[N[(t$95$2 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.4e+18], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * t$95$3), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.50000000000000001e-28

   1. Initial program 60.2%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in l around 0 51.5%

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

      1. unpow2 51.5%

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

      2. times-frac 53.4%

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

      3. unpow2 53.4%

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

      4. associate-*r/ 56.3%

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

   6. Simplified 56.3%

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

   if 9.50000000000000001e-28 < l < 7.4e18

   1. Initial program 40.9%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in U around 0 52.4%

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

      1. associate-*r* 52.4%

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

      2. *-commutative 52.4%

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

      3. +-commutative 52.4%

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

      4. mul-1-neg 52.4%

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

      5. unsub-neg 52.4%

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

      6. associate-+l- 52.4%

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

      7. cancel-sign-sub-inv 52.4%

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

      8. metadata-eval 52.4%

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

      9. +-commutative 52.4%

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

   6. Simplified 52.7%

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

   if 7.4e18 < l

   1. Initial program 41.4%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in l around 0 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. times-frac 57.0%

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

      5. associate-*r/ 57.0%

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

      6. metadata-eval 57.0%

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

   6. Simplified 57.0%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U* - U}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om} + \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 3: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om} + \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 3.9e-28)
     (sqrt
      (*
       t_1
       (+
        (+ t (* (/ (* l l) Om) -2.0))
        (* n (* (pow (/ l Om) 2.0) (- U* U))))))
     (if (<= l 3.5e+18)
       (sqrt
        (*
         (* 2.0 n)
         (*
          U
          (+ (* (/ n Om) (/ (* (* l l) U*) Om)) (fma -2.0 (* l (/ l Om)) t)))))
       (sqrt
        (*
         t_1
         (+ t (* (* l l) (- (* (/ n Om) (/ (- U* U) Om)) (/ 2.0 Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 3.9e-28) {
		tmp = sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	} else if (l <= 3.5e+18) {
		tmp = sqrt(((2.0 * n) * (U * (((n / Om) * (((l * l) * U_42_) / Om)) + fma(-2.0, (l * (l / Om)), t)))));
	} else {
		tmp = sqrt((t_1 * (t + ((l * l) * (((n / Om) * ((U_42_ - U) / Om)) - (2.0 / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 3.9e-28)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))));
	elseif (l <= 3.5e+18)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(Float64(n / Om) * Float64(Float64(Float64(l * l) * U_42_) / Om)) + fma(-2.0, Float64(l * Float64(l / Om)), t)))));
	else
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) - Float64(2.0 / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.9e-28], N[Sqrt[N[(t$95$1 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.5e+18], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.89999999999999999e-28

   1. Initial program 60.2%

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

   3. Simplified 59.1%

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

   if 3.89999999999999999e-28 < l < 3.5e18

   1. Initial program 40.9%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in U around 0 52.4%

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

      1. associate-*r* 52.4%

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

      2. *-commutative 52.4%

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

      3. +-commutative 52.4%

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

      4. mul-1-neg 52.4%

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

      5. unsub-neg 52.4%

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

      6. associate-+l- 52.4%

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

      7. cancel-sign-sub-inv 52.4%

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

      8. metadata-eval 52.4%

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

      9. +-commutative 52.4%

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

   6. Simplified 52.7%

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

   if 3.5e18 < l

   1. Initial program 41.4%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in l around 0 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. unpow2 53.9%

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

      4. times-frac 57.0%

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

      5. associate-*r/ 57.0%

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

      6. metadata-eval 57.0%

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

   6. Simplified 57.0%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om} + \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 4: 50.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -2 \cdot 10^{-107} \lor \neg \left(U* \leq 2.45 \cdot 10^{+51}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{\frac{n}{Om}}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -2e-107) (not (<= U* 2.45e+51)))
   (sqrt (* (* 2.0 (* n U)) (+ t (* l (* l (* (/ (/ n Om) Om) (- U* U)))))))
   (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -2e-107) || !(U_42_ <= 2.45e+51)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (l * (l * (((n / Om) / Om) * (U_42_ - U)))))));
	} else {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-2d-107)) .or. (.not. (u_42 <= 2.45d+51))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (l * (l * (((n / om) / om) * (u_42 - u)))))))
    else
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -2e-107) || !(U_42_ <= 2.45e+51)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (l * (l * (((n / Om) / Om) * (U_42_ - U)))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -2e-107) or not (U_42_ <= 2.45e+51):
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (l * (l * (((n / Om) / Om) * (U_42_ - U)))))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -2e-107) || !(U_42_ <= 2.45e+51))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) / Om) * Float64(U_42_ - U)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -2e-107) || ~((U_42_ <= 2.45e+51)))
		tmp = sqrt(((2.0 * (n * U)) * (t + (l * (l * (((n / Om) / Om) * (U_42_ - U)))))));
	else
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -2e-107], N[Not[LessEqual[U$42$, 2.45e+51]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -2e-107 or 2.44999999999999992e51 < U*

   1. Initial program 53.4%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in l around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. times-frac 54.9%

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

      5. associate-*r/ 54.9%

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

      6. metadata-eval 54.9%

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

   6. Simplified 54.9%

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

   7. Taylor expanded in n around -inf 47.0%

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

      1. unpow2 47.0%

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

      2. times-frac 49.9%

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

      3. *-commutative 49.9%

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

      4. associate-*l/ 52.4%

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

      5. associate-*l* 53.6%

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

      6. *-commutative 53.6%

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

      7. unpow2 53.6%

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

      8. associate-*l* 56.9%

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

      9. associate-*r/ 56.9%

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

      10. associate-/l* 56.9%

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

      11. associate-/r/ 57.6%

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

   9. Simplified 57.6%

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

   if -2e-107 < U* < 2.44999999999999992e51

   1. Initial program 55.9%

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

      1. associate-*l/ 58.7%

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

   3. Applied egg-rr 58.7%

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

   4. Taylor expanded in n around 0 60.2%

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

      1. *-commutative 60.2%

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

      2. cancel-sign-sub-inv 60.2%

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

      3. metadata-eval 60.2%

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

      4. *-commutative 60.2%

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

      5. unpow2 60.2%

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

      6. associate-*r/ 63.1%

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

   6. Simplified 63.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2 \cdot 10^{-107} \lor \neg \left(U* \leq 2.45 \cdot 10^{+51}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{\frac{n}{Om}}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 5: 49.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -2.6 \cdot 10^{-106} \lor \neg \left(U* \leq 10^{+50}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{\frac{\frac{Om}{\frac{U*}{Om}}}{\ell \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -2.6e-106) (not (<= U* 1e+50)))
   (sqrt (* (* 2.0 (* n U)) (+ t (/ n (/ (/ Om (/ U* Om)) (* l l))))))
   (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -2.6e-106) || !(U_42_ <= 1e+50)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (n / ((Om / (U_42_ / Om)) / (l * l))))));
	} else {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-2.6d-106)) .or. (.not. (u_42 <= 1d+50))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (n / ((om / (u_42 / om)) / (l * l))))))
    else
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -2.6e-106) || !(U_42_ <= 1e+50)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (n / ((Om / (U_42_ / Om)) / (l * l))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -2.6e-106) or not (U_42_ <= 1e+50):
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (n / ((Om / (U_42_ / Om)) / (l * l))))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -2.6e-106) || !(U_42_ <= 1e+50))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(n / Float64(Float64(Om / Float64(U_42_ / Om)) / Float64(l * l))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -2.6e-106) || ~((U_42_ <= 1e+50)))
		tmp = sqrt(((2.0 * (n * U)) * (t + (n / ((Om / (U_42_ / Om)) / (l * l))))));
	else
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -2.6e-106], N[Not[LessEqual[U$42$, 1e+50]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(n / N[(N[(Om / N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.6000000000000001e-106 or 1.0000000000000001e50 < U*

   1. Initial program 53.4%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in l around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. times-frac 54.9%

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

      5. associate-*r/ 54.9%

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

      6. metadata-eval 54.9%

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

   6. Simplified 54.9%

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

   7. Taylor expanded in U* around inf 47.0%

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

      1. mul-1-neg 47.0%

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

      2. associate-/l* 46.3%

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

      3. distribute-neg-frac 46.3%

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

      4. *-commutative 46.3%

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

      5. associate-/r* 51.0%

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

      6. unpow2 51.0%

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

      7. associate-/l* 52.4%

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

      8. unpow2 52.4%

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

   9. Simplified 52.4%

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

   if -2.6000000000000001e-106 < U* < 1.0000000000000001e50

   1. Initial program 55.9%

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

      1. associate-*l/ 58.7%

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

   3. Applied egg-rr 58.7%

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

   4. Taylor expanded in n around 0 60.2%

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

      1. *-commutative 60.2%

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

      2. cancel-sign-sub-inv 60.2%

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

      3. metadata-eval 60.2%

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

      4. *-commutative 60.2%

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

      5. unpow2 60.2%

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

      6. associate-*r/ 63.1%

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

   6. Simplified 63.1%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.6 \cdot 10^{-106} \lor \neg \left(U* \leq 10^{+50}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{n}{\frac{\frac{Om}{\frac{U*}{Om}}}{\ell \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 6: 42.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \frac{n}{\frac{\frac{Om \cdot Om}{U*}}{\ell \cdot \ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.1e-29)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (if (<= l 1.5e+172)
     (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))
     (sqrt (* 2.0 (* n (* U (/ n (/ (/ (* Om Om) U*) (* l l))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.1e-29) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = sqrt((2.0 * (n * (U * (n / (((Om * Om) / U_42_) / (l * l)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.1d-29) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else if (l <= 1.5d+172) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    else
        tmp = sqrt((2.0d0 * (n * (u * (n / (((om * om) / u_42) / (l * l)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.1e-29) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (n / (((Om * Om) / U_42_) / (l * l)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.1e-29:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	elif l <= 1.5e+172:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * (n / (((Om * Om) / U_42_) / (l * l)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.1e-29)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(n / Float64(Float64(Float64(Om * Om) / U_42_) / Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.1e-29)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	else
		tmp = sqrt((2.0 * (n * (U * (n / (((Om * Om) / U_42_) / (l * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.1e-29], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.5e+172], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(n / N[(N[(N[(Om * Om), $MachinePrecision] / U$42$), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.09999999999999995e-29

   1. Initial program 60.2%

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

      1. associate-*l/ 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 46.5%

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

      2. *-commutative 46.5%

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

      3. *-commutative 46.5%

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

   6. Applied egg-rr 46.5%

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

   if 1.09999999999999995e-29 < l < 1.5e172

   1. Initial program 57.8%

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

      1. associate-*l/ 59.6%

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

   3. Applied egg-rr 59.6%

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

   4. Taylor expanded in n around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. cancel-sign-sub-inv 55.5%

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

      3. metadata-eval 55.5%

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

      4. *-commutative 55.5%

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

      5. unpow2 55.5%

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

      6. associate-*r/ 57.3%

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

   6. Simplified 57.3%

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

   if 1.5e172 < l

   1. Initial program 9.6%

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

      1. associate-*l/ 13.1%

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

   3. Applied egg-rr 13.1%

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

   4. Taylor expanded in U* around inf 31.5%

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

      1. associate-*r* 31.5%

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

      2. unpow2 31.5%

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

      3. times-frac 35.5%

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

      4. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. *-un-lft-identity 35.5%

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

      2. associate-*l* 35.9%

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

      3. associate-/l* 35.9%

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

   8. Applied egg-rr 35.9%

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

      1. *-lft-identity 35.9%

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

      2. associate-*l* 35.9%

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

      3. unpow2 35.9%

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

      4. associate-/l* 35.9%

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

      5. unpow2 35.9%

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

      6. times-frac 31.7%

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

      7. unpow2 31.7%

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

      8. associate-*r* 31.7%

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

      9. unpow2 31.7%

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

      10. associate-/l* 31.7%

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

      11. unpow2 31.7%

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

      12. times-frac 35.6%

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

      13. associate-*l/ 35.3%

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

      14. unpow2 35.3%

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

      15. associate-*r/ 31.5%

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

   10. Simplified 31.5%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \frac{n}{\frac{\frac{Om \cdot Om}{U*}}{\ell \cdot \ell}}\right)\right)}\\ \end{array} \]

Alternative 7: 42.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-30}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 9.5e-30)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (if (<= l 1.5e+172)
     (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))
     (sqrt (* 2.0 (* (/ (* (* n l) (* n l)) Om) (/ (* U U*) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 9.5e-30) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 9.5d-30) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else if (l <= 1.5d+172) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    else
        tmp = sqrt((2.0d0 * ((((n * l) * (n * l)) / om) * ((u * u_42) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 9.5e-30) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = Math.sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 9.5e-30:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	elif l <= 1.5e+172:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	else:
		tmp = math.sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 9.5e-30)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(n * l) * Float64(n * l)) / Om) * Float64(Float64(U * U_42_) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 9.5e-30)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	else
		tmp = sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 9.5e-30], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.5e+172], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.49999999999999939e-30

   1. Initial program 60.2%

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

      1. associate-*l/ 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 46.5%

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

      2. *-commutative 46.5%

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

      3. *-commutative 46.5%

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

   6. Applied egg-rr 46.5%

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

   if 9.49999999999999939e-30 < l < 1.5e172

   1. Initial program 57.8%

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

      1. associate-*l/ 59.6%

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

   3. Applied egg-rr 59.6%

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

   4. Taylor expanded in n around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. cancel-sign-sub-inv 55.5%

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

      3. metadata-eval 55.5%

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

      4. *-commutative 55.5%

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

      5. unpow2 55.5%

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

      6. associate-*r/ 57.3%

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

   6. Simplified 57.3%

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

   if 1.5e172 < l

   1. Initial program 9.6%

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

      1. associate-*l/ 13.1%

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

   3. Applied egg-rr 13.1%

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

   4. Taylor expanded in U* around inf 26.8%

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

      1. associate-*r* 26.8%

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

      2. unpow2 26.8%

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

      3. times-frac 31.2%

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

      4. unpow2 31.2%

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

      5. unpow2 31.2%

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

      6. unswap-sqr 35.7%

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

   6. Simplified 35.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-30}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \end{array} \]

Alternative 8: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-30}:\\ \;\;\;\;{\left(t_1 \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l 7.8e-30)
     (pow (* t_1 t) 0.5)
     (if (<= l 1.5e+172)
       (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))
       (sqrt (* t_1 (* (/ U* Om) (/ (* n (* l l)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l <= 7.8e-30) {
		tmp = pow((t_1 * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = sqrt((t_1 * ((U_42_ / Om) * ((n * (l * l)) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (l <= 7.8d-30) then
        tmp = (t_1 * t) ** 0.5d0
    else if (l <= 1.5d+172) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    else
        tmp = sqrt((t_1 * ((u_42 / om) * ((n * (l * l)) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l <= 7.8e-30) {
		tmp = Math.pow((t_1 * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = Math.sqrt((t_1 * ((U_42_ / Om) * ((n * (l * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if l <= 7.8e-30:
		tmp = math.pow((t_1 * t), 0.5)
	elif l <= 1.5e+172:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	else:
		tmp = math.sqrt((t_1 * ((U_42_ / Om) * ((n * (l * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l <= 7.8e-30)
		tmp = Float64(t_1 * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	else
		tmp = sqrt(Float64(t_1 * Float64(Float64(U_42_ / Om) * Float64(Float64(n * Float64(l * l)) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (l <= 7.8e-30)
		tmp = (t_1 * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	else
		tmp = sqrt((t_1 * ((U_42_ / Om) * ((n * (l * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l, 7.8e-30], N[Power[N[(t$95$1 * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.5e+172], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.8000000000000007e-30

   1. Initial program 60.2%

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

      1. associate-*l/ 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 46.5%

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

      2. *-commutative 46.5%

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

      3. *-commutative 46.5%

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

   6. Applied egg-rr 46.5%

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

   if 7.8000000000000007e-30 < l < 1.5e172

   1. Initial program 57.8%

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

      1. associate-*l/ 59.6%

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

   3. Applied egg-rr 59.6%

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

   4. Taylor expanded in n around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. cancel-sign-sub-inv 55.5%

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

      3. metadata-eval 55.5%

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

      4. *-commutative 55.5%

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

      5. unpow2 55.5%

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

      6. associate-*r/ 57.3%

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

   6. Simplified 57.3%

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

   if 1.5e172 < l

   1. Initial program 9.6%

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

      1. associate-*l/ 13.1%

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

   3. Applied egg-rr 13.1%

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

   4. Taylor expanded in U* around inf 31.5%

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

      1. associate-*r* 31.5%

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

      2. unpow2 31.5%

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

      3. times-frac 35.5%

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

      4. unpow2 35.5%

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

   6. Simplified 35.5%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-30}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \]

Alternative 9: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-30}:\\ \;\;\;\;{\left(t_1 \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \frac{U* \cdot \frac{n}{\frac{Om}{\ell \cdot \ell}}}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l 8.2e-30)
     (pow (* t_1 t) 0.5)
     (if (<= l 1.5e+172)
       (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))
       (sqrt (* t_1 (/ (* U* (/ n (/ Om (* l l)))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l <= 8.2e-30) {
		tmp = pow((t_1 * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = sqrt((t_1 * ((U_42_ * (n / (Om / (l * l)))) / Om)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (l <= 8.2d-30) then
        tmp = (t_1 * t) ** 0.5d0
    else if (l <= 1.5d+172) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    else
        tmp = sqrt((t_1 * ((u_42 * (n / (om / (l * l)))) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l <= 8.2e-30) {
		tmp = Math.pow((t_1 * t), 0.5);
	} else if (l <= 1.5e+172) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	} else {
		tmp = Math.sqrt((t_1 * ((U_42_ * (n / (Om / (l * l)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if l <= 8.2e-30:
		tmp = math.pow((t_1 * t), 0.5)
	elif l <= 1.5e+172:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	else:
		tmp = math.sqrt((t_1 * ((U_42_ * (n / (Om / (l * l)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l <= 8.2e-30)
		tmp = Float64(t_1 * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	else
		tmp = sqrt(Float64(t_1 * Float64(Float64(U_42_ * Float64(n / Float64(Om / Float64(l * l)))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (l <= 8.2e-30)
		tmp = (t_1 * t) ^ 0.5;
	elseif (l <= 1.5e+172)
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	else
		tmp = sqrt((t_1 * ((U_42_ * (n / (Om / (l * l)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l, 8.2e-30], N[Power[N[(t$95$1 * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.5e+172], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(N[(U$42$ * N[(n / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.2000000000000007e-30

   1. Initial program 60.2%

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

      1. associate-*l/ 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 46.5%

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

      2. *-commutative 46.5%

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

      3. *-commutative 46.5%

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

   6. Applied egg-rr 46.5%

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

   if 8.2000000000000007e-30 < l < 1.5e172

   1. Initial program 57.8%

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

      1. associate-*l/ 59.6%

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

   3. Applied egg-rr 59.6%

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

   4. Taylor expanded in n around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. cancel-sign-sub-inv 55.5%

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

      3. metadata-eval 55.5%

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

      4. *-commutative 55.5%

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

      5. unpow2 55.5%

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

      6. associate-*r/ 57.3%

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

   6. Simplified 57.3%

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

   if 1.5e172 < l

   1. Initial program 9.6%

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

      1. associate-*l/ 13.1%

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

   3. Applied egg-rr 13.1%

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

   4. Taylor expanded in U* around inf 31.5%

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

      1. associate-*r* 31.5%

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

      2. unpow2 31.5%

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

      3. times-frac 35.5%

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

      4. unpow2 35.5%

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

   6. Simplified 35.5%

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

      1. associate-*r/ 36.2%

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

      2. associate-/l* 36.2%

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

   8. Applied egg-rr 36.2%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-30}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \frac{n}{\frac{Om}{\ell \cdot \ell}}}{Om}}\\ \end{array} \]

Alternative 10: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-29}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.18e-29)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (sqrt (* 2.0 (* n (* U (+ t (* (/ (* l l) Om) -2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.18e-29) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = sqrt((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.18d-29) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (n * (u * (t + (((l * l) / om) * (-2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.18e-29) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.18e-29:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = math.sqrt((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.18e-29)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.18e-29)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.18e-29], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.17999999999999996e-29

   1. Initial program 60.2%

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

      1. associate-*l/ 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 46.5%

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

      2. *-commutative 46.5%

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

      3. *-commutative 46.5%

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

   6. Applied egg-rr 46.5%

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

   if 1.17999999999999996e-29 < l

   1. Initial program 41.3%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in n around 0 40.2%

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

      1. cancel-sign-sub-inv 40.2%

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

      2. metadata-eval 40.2%

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

      3. *-commutative 40.2%

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

      4. unpow2 40.2%

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

   6. Simplified 40.2%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-29}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 11: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-29}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.35e-29)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (sqrt (* 2.0 (* n (* U (+ t (* (* l (/ l Om)) -2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.35e-29) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.35d-29) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (n * (u * (t + ((l * (l / om)) * (-2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.35e-29) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.35e-29:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = math.sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.35e-29)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.35e-29)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt((2.0 * (n * (U * (t + ((l * (l / Om)) * -2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.35e-29], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.35000000000000011e-29

   1. Initial program 60.2%

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

      1. associate-*l/ 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in t around inf 43.6%

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

      1. pow1/2 46.5%

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

      2. *-commutative 46.5%

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

      3. *-commutative 46.5%

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

   6. Applied egg-rr 46.5%

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

   if 1.35000000000000011e-29 < l

   1. Initial program 41.3%

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

      1. associate-*l/ 43.7%

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

   3. Applied egg-rr 43.7%

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

   4. Taylor expanded in n around 0 40.2%

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

      1. *-commutative 40.2%

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

      2. cancel-sign-sub-inv 40.2%

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

      3. metadata-eval 40.2%

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

      4. *-commutative 40.2%

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

      5. unpow2 40.2%

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

      6. associate-*r/ 42.6%

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

   6. Simplified 42.6%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-29}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 12: 38.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+107}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.5e+107)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (sqrt (* -2.0 (* n (* 2.0 (/ (* l l) (/ Om U))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.5e+107) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = sqrt((-2.0 * (n * (2.0 * ((l * l) / (Om / U))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 3.5d+107) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = sqrt(((-2.0d0) * (n * (2.0d0 * ((l * l) / (om / u))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.5e+107) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (n * (2.0 * ((l * l) / (Om / U))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.5e+107:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (n * (2.0 * ((l * l) / (Om / U))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.5e+107)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(n * Float64(2.0 * Float64(Float64(l * l) / Float64(Om / U))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.5e+107)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (n * (2.0 * ((l * l) / (Om / U))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.5e+107], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(n * N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.4999999999999997e107

   1. Initial program 60.0%

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

      1. associate-*l/ 61.4%

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

   3. Applied egg-rr 61.4%

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

   4. Taylor expanded in t around inf 43.3%

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

      1. pow1/2 45.7%

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

      2. *-commutative 45.7%

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

      3. *-commutative 45.7%

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

   6. Applied egg-rr 45.7%

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

   if 3.4999999999999997e107 < l

   1. Initial program 25.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in l around inf 41.0%

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

      1. unpow2 41.0%

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

      2. associate-*r/ 41.0%

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

      3. metadata-eval 41.0%

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

      4. *-commutative 41.0%

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

      5. unpow2 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in U around 0 43.0%

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

      1. unpow2 43.0%

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

      2. *-commutative 43.0%

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

      3. fma-def 43.0%

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

      4. unpow2 43.0%

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

      5. associate-*r/ 43.0%

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

      6. metadata-eval 43.0%

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

   9. Simplified 43.0%

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

   10. Taylor expanded in n around 0 20.1%

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

       1. associate-/l* 21.5%

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

       2. unpow2 21.5%

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

   12. Simplified 21.5%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+107}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)\right)}\\ \end{array} \]

Alternative 13: 36.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-144}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.8e-144)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.8e-144) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-2.8d-144)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.8e-144) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -2.8e-144:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -2.8e-144)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -2.8e-144)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -2.8e-144], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.79999999999999998e-144

   1. Initial program 54.0%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in t around inf 35.9%

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

      1. associate-*r* 35.9%

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

      2. *-commutative 35.9%

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

   6. Simplified 35.9%

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

      1. pow1/2 39.4%

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

      2. associate-*l* 39.4%

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

   8. Applied egg-rr 39.4%

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

   if -2.79999999999999998e-144 < n

   1. Initial program 54.6%

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

   3. Simplified 53.3%

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

      1. add-cube-cbrt 53.2%

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

      2. pow3 53.2%

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

      3. *-commutative 53.2%

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

      4. associate-*r* 56.3%

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

      5. *-commutative 56.3%

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

      6. *-commutative 56.3%

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

   5. Applied egg-rr 56.3%

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

   6. Taylor expanded in n around inf 36.4%

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

      1. associate-*r* 43.8%

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

   8. Simplified 43.8%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-144}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 14: 36.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.95e-143)
   (sqrt (* (* 2.0 n) (* U t)))
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.95e-143) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-1.95d-143)) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.95e-143) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -1.95e-143:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.95e-143)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -1.95e-143)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.95e-143], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.95000000000000002e-143

   1. Initial program 54.0%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in t around inf 35.9%

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

      1. associate-*r* 35.9%

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

      2. *-commutative 35.9%

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

   6. Simplified 35.9%

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

   if -1.95000000000000002e-143 < n

   1. Initial program 54.6%

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

   3. Simplified 53.3%

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

      1. add-cube-cbrt 53.2%

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

      2. pow3 53.2%

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

      3. *-commutative 53.2%

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

      4. associate-*r* 56.3%

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

      5. *-commutative 56.3%

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

      6. *-commutative 56.3%

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

   5. Applied egg-rr 56.3%

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

   6. Taylor expanded in n around inf 36.4%

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

      1. associate-*r* 43.8%

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

   8. Simplified 43.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15: 37.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. associate-*l/ 56.3%

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

3. Applied egg-rr 56.3%

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

4. Taylor expanded in t around inf 38.8%

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

   1. pow1/2 41.6%

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

   2. *-commutative 41.6%

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

   3. *-commutative 41.6%

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

6. Applied egg-rr 41.6%

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

7. Final simplification 41.6%

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

Alternative 16: 36.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Simplified 53.4%

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

   1. add-cube-cbrt 53.3%

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

   2. pow3 53.3%

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

   3. *-commutative 53.3%

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

   4. associate-*r* 56.2%

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

   5. *-commutative 56.2%

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

   6. *-commutative 56.2%

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

5. Applied egg-rr 56.2%

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

6. Taylor expanded in n around inf 36.2%

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

   1. associate-*r* 39.1%

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

8. Simplified 39.1%

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

9. Final simplification 39.1%

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

Reproduce

herbie shell --seed 2023273 
(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*))))))