Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 63.2%

Time: 34.0s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_1 \cdot \left(U - U*\right)\right)\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t_2 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{n \cdot n}{\frac{\frac{Om}{\ell}}{\ell} \cdot \frac{Om}{U \cdot \left(U - U*\right)}}, \left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t + \left(t_1 \cdot \left(U* - U\right) - 2 \cdot t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot t_3\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U U*)))))
        (t_3 (* l (/ l Om))))
   (if (<= t_2 5e-324)
     (sqrt
      (fma
       -2.0
       (/ (* n n) (* (/ (/ Om l) l) (/ Om (* U (- U U*)))))
       (* (* 2.0 n) (* U (fma l (* (/ l Om) -2.0) t)))))
     (if (<= t_2 INFINITY)
       (* (sqrt 2.0) (sqrt (* (* n U) (+ t (- (* t_1 (- U* U)) (* 2.0 t_3))))))
       (pow (* 2.0 (* n (* U (+ t (* -2.0 t_3))))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - (t_1 * (U - U_42_)));
	double t_3 = l * (l / Om);
	double tmp;
	if (t_2 <= 5e-324) {
		tmp = sqrt(fma(-2.0, ((n * n) / (((Om / l) / l) * (Om / (U * (U - U_42_))))), ((2.0 * n) * (U * fma(l, ((l / Om) * -2.0), t)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * sqrt(((n * U) * (t + ((t_1 * (U_42_ - U)) - (2.0 * t_3)))));
	} else {
		tmp = pow((2.0 * (n * (U * (t + (-2.0 * t_3))))), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(t_1 * Float64(U - U_42_))))
	t_3 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (t_2 <= 5e-324)
		tmp = sqrt(fma(-2.0, Float64(Float64(n * n) / Float64(Float64(Float64(Om / l) / l) * Float64(Om / Float64(U * Float64(U - U_42_))))), Float64(Float64(2.0 * n) * Float64(U * fma(l, Float64(Float64(l / Om) * -2.0), t)))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(Float64(n * U) * Float64(t + Float64(Float64(t_1 * Float64(U_42_ - U)) - Float64(2.0 * t_3))))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * t_3))))) ^ 0.5;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-324], N[Sqrt[N[(-2.0 * N[(N[(n * n), $MachinePrecision] / N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] * N[(Om / N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.94066e-324

   1. Initial program 10.2%

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

   3. Simplified 10.2%

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

   4. Taylor expanded in n around 0 39.6%

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

      1. fma-def 39.6%

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

      2. associate-/l* 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. times-frac 43.8%

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

      6. unpow2 43.8%

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

      7. associate-/r* 43.9%

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

      8. *-commutative 43.9%

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

      9. associate-*r* 44.4%

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

      10. cancel-sign-sub-inv 44.4%

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

      11. metadata-eval 44.4%

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

      12. *-commutative 44.4%

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

      13. +-commutative 44.4%

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

   6. Simplified 44.4%

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

   if 4.94066e-324 < (*.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 64.6%

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

   3. Simplified 66.5%

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

      1. pow1/2 66.5%

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

      2. associate-*l* 66.4%

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

      3. unpow-prod-down 67.3%

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

      4. pow1/2 67.3%

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

      5. *-commutative 67.3%

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

      6. associate-*r* 70.7%

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

      7. *-commutative 70.7%

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

      8. *-commutative 70.7%

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

   5. Applied egg-rr 70.7%

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

      1. unpow1/2 70.7%

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

   7. Simplified 70.7%

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

      1. fma-udef 70.7%

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

   9. Applied egg-rr 70.7%

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

   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 3.0%

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

      1. pow1/2 3.0%

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

      2. fma-udef 3.0%

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

      3. associate-*l/ 0.1%

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

      4. associate-*r* 0.0%

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

      5. *-commutative 0.0%

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

      6. associate--l- 0.0%

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

      7. cancel-sign-sub-inv 0.0%

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

      8. metadata-eval 0.0%

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

      9. associate-*r* 0.1%

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

   5. Applied egg-rr 2.9%

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

   6. Taylor expanded in n around 0 36.6%

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

      1. *-commutative 36.6%

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

      2. cancel-sign-sub-inv 36.6%

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

      3. metadata-eval 36.6%

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

      4. *-commutative 36.6%

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

      5. unpow2 36.6%

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

      6. associate-*r/ 44.1%

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

   8. Simplified 44.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{n \cdot n}{\frac{\frac{Om}{\ell}}{\ell} \cdot \frac{Om}{U \cdot \left(U - U*\right)}}, \left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 2: 61.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{Om}{\ell}}{\ell}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t_2 \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{t_1}{U}}\right)}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{t_1 \cdot \frac{Om}{U*}} - \frac{2}{t_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{n \cdot \ell}{\frac{\frac{Om}{U \cdot U*}}{n \cdot \ell}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (/ Om l) l))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 4e-101)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* n -4.0) (/ t_1 U))))
     (if (<= t_2 4e+120)
       t_2
       (if (<= t_2 INFINITY)
         (sqrt
          (* (* 2.0 (* n U)) (+ t (- (/ n (* t_1 (/ Om U*))) (/ 2.0 t_1)))))
         (sqrt (* (/ 2.0 Om) (/ (* n l) (/ (/ Om (* U U*)) (* n l))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (Om / l) / l;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 4e-101) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((n * -4.0) / (t_1 / U))));
	} else if (t_2 <= 4e+120) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / (t_1 * (Om / U_42_))) - (2.0 / t_1)))));
	} else {
		tmp = sqrt(((2.0 / Om) * ((n * l) / ((Om / (U * U_42_)) / (n * l)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Om / l) / l)
	t_2 = 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_)))))
	tmp = 0.0
	if (t_2 <= 4e-101)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(n * -4.0) / Float64(t_1 / U))));
	elseif (t_2 <= 4e+120)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n / Float64(t_1 * Float64(Om / U_42_))) - Float64(2.0 / t_1)))));
	else
		tmp = sqrt(Float64(Float64(2.0 / Om) * Float64(Float64(n * l) / Float64(Float64(Om / Float64(U * U_42_)) / Float64(n * l)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 4e-101], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(n * -4.0), $MachinePrecision] / N[(t$95$1 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+120], t$95$2, If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n / N[(t$95$1 * N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / N[(N[(Om / N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 4.00000000000000021e-101

   1. Initial program 25.3%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in Om around inf 47.0%

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

      1. fma-def 47.0%

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

      2. associate-*r* 49.2%

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

      3. associate-/l* 48.4%

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

      4. associate-*r/ 48.4%

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

      5. associate-/r* 48.4%

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

      6. unpow2 48.4%

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

      7. associate-/r* 48.4%

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

   6. Simplified 48.4%

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

   if 4.00000000000000021e-101 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 3.9999999999999999e120

   1. Initial program 99.8%

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

   if 3.9999999999999999e120 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.9%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in U around 0 30.7%

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

      1. +-commutative 30.7%

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

      2. mul-1-neg 30.7%

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

      3. unsub-neg 30.7%

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

      4. associate-*r/ 30.7%

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

      5. associate-/l* 30.7%

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

      6. unpow2 30.7%

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

      7. associate-/r* 30.7%

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

      8. associate-/l* 30.7%

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

      9. unpow2 30.7%

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

      10. times-frac 33.1%

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

      11. unpow2 33.1%

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

      12. associate-/r* 44.1%

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

   6. Simplified 44.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.8%

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

   4. Taylor expanded in U* around inf 32.4%

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

      1. associate-*r/ 32.4%

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

      2. unpow2 32.4%

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

      3. times-frac 35.3%

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

      4. associate-*r* 35.3%

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

      5. unpow2 35.3%

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

      6. unpow2 35.3%

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

   6. Simplified 35.3%

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

      1. *-un-lft-identity 35.3%

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

      2. associate-/l* 34.8%

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

      3. unswap-sqr 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. *-lft-identity 40.8%

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

      2. associate-/l* 43.2%

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

   10. Simplified 43.2%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 4 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{\frac{Om}{\ell}}{\ell} \cdot \frac{Om}{U*}} - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{n \cdot \ell}{\frac{\frac{Om}{U \cdot U*}}{n \cdot \ell}}}\\ \end{array} \]

Alternative 3: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_1 \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t_2 \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t + \left(t_1 \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{n \cdot \ell}{\frac{\frac{Om}{U \cdot U*}}{n \cdot \ell}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U U*)))))))
   (if (<= t_2 4e-101)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* n -4.0) (/ (/ (/ Om l) l) U))))
     (if (<= t_2 INFINITY)
       (*
        (sqrt 2.0)
        (sqrt (* (* n U) (+ t (- (* t_1 (- U* U)) (* 2.0 (* l (/ l Om))))))))
       (sqrt (* (/ 2.0 Om) (/ (* n l) (/ (/ Om (* U U*)) (* n l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - (t_1 * (U - U_42_)))));
	double tmp;
	if (t_2 <= 4e-101) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((n * -4.0) / (((Om / l) / l) / U))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * sqrt(((n * U) * (t + ((t_1 * (U_42_ - U)) - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(((2.0 / Om) * ((n * l) / ((Om / (U * U_42_)) / (n * l)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(t_1 * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 4e-101)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(n * -4.0) / Float64(Float64(Float64(Om / l) / l) / U))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(Float64(n * U) * Float64(t + Float64(Float64(t_1 * Float64(U_42_ - U)) - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 / Om) * Float64(Float64(n * l) / Float64(Float64(Om / Float64(U * U_42_)) / Float64(n * l)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4e-101], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(n * -4.0), $MachinePrecision] / N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / N[(N[(Om / N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 4.00000000000000021e-101

   1. Initial program 25.3%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in Om around inf 47.0%

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

      1. fma-def 47.0%

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

      2. associate-*r* 49.2%

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

      3. associate-/l* 48.4%

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

      4. associate-*r/ 48.4%

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

      5. associate-/r* 48.4%

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

      6. unpow2 48.4%

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

      7. associate-/r* 48.4%

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

   6. Simplified 48.4%

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

   if 4.00000000000000021e-101 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.7%

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

   3. Simplified 65.7%

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

      1. pow1/2 65.7%

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

      2. associate-*l* 65.7%

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

      3. unpow-prod-down 66.6%

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

      4. pow1/2 66.6%

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

      5. *-commutative 66.6%

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

      6. associate-*r* 70.2%

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

      7. *-commutative 70.2%

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

      8. *-commutative 70.2%

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

   5. Applied egg-rr 70.2%

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

      1. unpow1/2 70.2%

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

   7. Simplified 70.2%

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

      1. fma-udef 70.2%

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

   9. Applied egg-rr 70.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.8%

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

   4. Taylor expanded in U* around inf 32.4%

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

      1. associate-*r/ 32.4%

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

      2. unpow2 32.4%

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

      3. times-frac 35.3%

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

      4. associate-*r* 35.3%

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

      5. unpow2 35.3%

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

      6. unpow2 35.3%

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

   6. Simplified 35.3%

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

      1. *-un-lft-identity 35.3%

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

      2. associate-/l* 34.8%

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

      3. unswap-sqr 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. *-lft-identity 40.8%

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

      2. associate-/l* 43.2%

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

   10. Simplified 43.2%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{n \cdot \ell}{\frac{\frac{Om}{U \cdot U*}}{n \cdot \ell}}}\\ \end{array} \]

Alternative 4: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(U - U*\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{n \cdot \ell}{\frac{\frac{Om}{U \cdot U*}}{n \cdot \ell}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 4e-101)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* n -4.0) (/ (/ (/ Om l) l) U))))
     (if (<= t_2 INFINITY)
       (pow
        (* (* 2.0 (* n U)) (- t (fma 2.0 (* l (/ l Om)) (* t_1 (- U U*)))))
        0.5)
       (sqrt (* (/ 2.0 Om) (/ (* n l) (/ (/ Om (* U U*)) (* n l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 4e-101) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((n * -4.0) / (((Om / l) / l) / U))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow(((2.0 * (n * U)) * (t - fma(2.0, (l * (l / Om)), (t_1 * (U - U_42_))))), 0.5);
	} else {
		tmp = sqrt(((2.0 / Om) * ((n * l) / ((Om / (U * U_42_)) / (n * l)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 4e-101)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(n * -4.0) / Float64(Float64(Float64(Om / l) / l) / U))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(U - U_42_))))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 / Om) * Float64(Float64(n * l) / Float64(Float64(Om / Float64(U * U_42_)) / Float64(n * l)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4e-101], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(n * -4.0), $MachinePrecision] / N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / N[(N[(Om / N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 4.00000000000000021e-101

   1. Initial program 25.3%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in Om around inf 47.0%

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

      1. fma-def 47.0%

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

      2. associate-*r* 49.2%

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

      3. associate-/l* 48.4%

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

      4. associate-*r/ 48.4%

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

      5. associate-/r* 48.4%

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

      6. unpow2 48.4%

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

      7. associate-/r* 48.4%

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

   6. Simplified 48.4%

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

   if 4.00000000000000021e-101 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.7%

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

   3. Simplified 65.7%

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

      1. pow1/2 65.7%

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

      2. fma-udef 65.7%

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

      3. associate-*l/ 60.1%

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

      4. associate-*r* 63.7%

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

      5. *-commutative 63.7%

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

      6. associate--l- 63.7%

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

      7. cancel-sign-sub-inv 63.7%

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

      8. metadata-eval 63.7%

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

      9. associate-*r* 63.2%

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

   5. Applied egg-rr 69.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.8%

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

   4. Taylor expanded in U* around inf 32.4%

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

      1. associate-*r/ 32.4%

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

      2. unpow2 32.4%

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

      3. times-frac 35.3%

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

      4. associate-*r* 35.3%

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

      5. unpow2 35.3%

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

      6. unpow2 35.3%

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

   6. Simplified 35.3%

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

      1. *-un-lft-identity 35.3%

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

      2. associate-/l* 34.8%

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

      3. unswap-sqr 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. *-lft-identity 40.8%

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

      2. associate-/l* 43.2%

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

   10. Simplified 43.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{Om} \cdot \frac{n \cdot \ell}{\frac{\frac{Om}{U \cdot U*}}{n \cdot \ell}}}\\ \end{array} \]

Alternative 5: 53.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.1e+44)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ (+ t (* (/ (* l l) Om) -2.0)) (* n (* (pow (/ l Om) 2.0) (- U* U))))))
   (pow (* 2.0 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.1e+44) {
		tmp = sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	} else {
		tmp = pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.1d+44) then
        tmp = sqrt(((2.0d0 * (n * u)) * ((t + (((l * l) / om) * (-2.0d0))) + (n * (((l / om) ** 2.0d0) * (u_42 - u))))))
    else
        tmp = (2.0d0 * (n * (u * (t + ((-2.0d0) * (l * (l / om))))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.1e+44) {
		tmp = Math.sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U))))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.1e+44:
		tmp = math.sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U))))))
	else:
		tmp = math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.1e+44)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.1e+44)
		tmp = sqrt(((2.0 * (n * U)) * ((t + (((l * l) / Om) * -2.0)) + (n * (((l / Om) ^ 2.0) * (U_42_ - U))))));
	else
		tmp = (2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.1e+44], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * 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], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.09999999999999998e44

   1. Initial program 52.4%

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

   3. Simplified 51.9%

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

   if 1.09999999999999998e44 < l

   1. Initial program 22.6%

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

   3. Simplified 37.2%

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

      1. pow1/2 37.2%

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

      2. fma-udef 37.2%

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

      3. associate-*l/ 22.7%

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

      4. associate-*r* 22.6%

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

      5. *-commutative 22.6%

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

      6. associate--l- 22.6%

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

      7. cancel-sign-sub-inv 22.6%

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

      8. metadata-eval 22.6%

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

      9. associate-*r* 22.7%

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

   5. Applied egg-rr 37.1%

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

   6. Taylor expanded in n around 0 37.8%

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

      1. *-commutative 37.8%

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

      2. cancel-sign-sub-inv 37.8%

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

      3. metadata-eval 37.8%

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

      4. *-commutative 37.8%

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

      5. unpow2 37.8%

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

      6. associate-*r/ 45.8%

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

   8. Simplified 45.8%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 6: 46.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{Om}{\ell}}{\ell}\\ \mathbf{if}\;\ell \leq 3.05 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{t_1 \cdot \frac{Om}{U*}} - \frac{2}{t_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (/ Om l) l)))
   (if (<= l 3.05e-174)
     (sqrt (* n (* t (* 2.0 U))))
     (if (<= l 3e+44)
       (sqrt (* (* 2.0 (* n U)) (+ t (- (/ n (* t_1 (/ Om U*))) (/ 2.0 t_1)))))
       (pow (* 2.0 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (Om / l) / l;
	double tmp;
	if (l <= 3.05e-174) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} else if (l <= 3e+44) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / (t_1 * (Om / U_42_))) - (2.0 / t_1)))));
	} else {
		tmp = pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (om / l) / l
    if (l <= 3.05d-174) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    else if (l <= 3d+44) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n / (t_1 * (om / u_42))) - (2.0d0 / t_1)))))
    else
        tmp = (2.0d0 * (n * (u * (t + ((-2.0d0) * (l * (l / om))))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (Om / l) / l;
	double tmp;
	if (l <= 3.05e-174) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else if (l <= 3e+44) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n / (t_1 * (Om / U_42_))) - (2.0 / t_1)))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (Om / l) / l
	tmp = 0
	if l <= 3.05e-174:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	elif l <= 3e+44:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n / (t_1 * (Om / U_42_))) - (2.0 / t_1)))))
	else:
		tmp = math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Om / l) / l)
	tmp = 0.0
	if (l <= 3.05e-174)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	elseif (l <= 3e+44)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n / Float64(t_1 * Float64(Om / U_42_))) - Float64(2.0 / t_1)))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (Om / l) / l;
	tmp = 0.0;
	if (l <= 3.05e-174)
		tmp = sqrt((n * (t * (2.0 * U))));
	elseif (l <= 3e+44)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n / (t_1 * (Om / U_42_))) - (2.0 / t_1)))));
	else
		tmp = (2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[l, 3.05e-174], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3e+44], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n / N[(t$95$1 * N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.04999999999999982e-174

   1. Initial program 50.0%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in t around inf 40.9%

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

      1. associate-*r* 39.0%

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

   6. Simplified 39.0%

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

      1. associate-*l* 40.9%

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

      2. *-commutative 40.9%

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

      3. expm1-log1p-u 39.7%

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

      4. expm1-udef 22.1%

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

   8. Applied egg-rr 22.1%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 40.9%

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

      3. *-commutative 40.9%

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

      4. associate-*l* 41.0%

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

      5. *-commutative 41.0%

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

      6. associate-*r* 41.0%

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

   10. Simplified 41.0%

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

   if 3.04999999999999982e-174 < l < 2.99999999999999987e44

   1. Initial program 59.7%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in U around 0 59.7%

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

      1. +-commutative 59.7%

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

      2. mul-1-neg 59.7%

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

      3. unsub-neg 59.7%

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

      4. associate-*r/ 59.7%

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

      5. associate-/l* 59.7%

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

      6. unpow2 59.7%

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

      7. associate-/r* 59.7%

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

      8. associate-/l* 61.7%

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

      9. unpow2 61.7%

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

      10. times-frac 59.8%

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

      11. unpow2 59.8%

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

      12. associate-/r* 61.7%

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

   6. Simplified 61.7%

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

   if 2.99999999999999987e44 < l

   1. Initial program 22.6%

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

   3. Simplified 37.2%

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

      1. pow1/2 37.2%

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

      2. fma-udef 37.2%

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

      3. associate-*l/ 22.7%

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

      4. associate-*r* 22.6%

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

      5. *-commutative 22.6%

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

      6. associate--l- 22.6%

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

      7. cancel-sign-sub-inv 22.6%

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

      8. metadata-eval 22.6%

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

      9. associate-*r* 22.7%

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

   5. Applied egg-rr 37.1%

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

   6. Taylor expanded in n around 0 37.8%

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

      1. *-commutative 37.8%

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

      2. cancel-sign-sub-inv 37.8%

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

      3. metadata-eval 37.8%

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

      4. *-commutative 37.8%

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

      5. unpow2 37.8%

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

      6. associate-*r/ 45.8%

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

   8. Simplified 45.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.05 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\frac{n}{\frac{\frac{Om}{\ell}}{\ell} \cdot \frac{Om}{U*}} - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7: 45.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-22}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.25e-163)
   (sqrt (* n (* t (* 2.0 U))))
   (if (<= l 7.2e-22)
     (pow
      (* (* 2.0 (* n U)) (- t (* (/ n Om) (/ (* (* l l) (- U U*)) Om))))
      0.5)
     (pow (* 2.0 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))) 0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.25e-163) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} else if (l <= 7.2e-22) {
		tmp = pow(((2.0 * (n * U)) * (t - ((n / Om) * (((l * l) * (U - U_42_)) / Om)))), 0.5);
	} else {
		tmp = pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.25d-163) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    else if (l <= 7.2d-22) then
        tmp = ((2.0d0 * (n * u)) * (t - ((n / om) * (((l * l) * (u - u_42)) / om)))) ** 0.5d0
    else
        tmp = (2.0d0 * (n * (u * (t + ((-2.0d0) * (l * (l / om))))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.25e-163) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else if (l <= 7.2e-22) {
		tmp = Math.pow(((2.0 * (n * U)) * (t - ((n / Om) * (((l * l) * (U - U_42_)) / Om)))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.25e-163:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	elif l <= 7.2e-22:
		tmp = math.pow(((2.0 * (n * U)) * (t - ((n / Om) * (((l * l) * (U - U_42_)) / Om)))), 0.5)
	else:
		tmp = math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.25e-163)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	elseif (l <= 7.2e-22)
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(n / Om) * Float64(Float64(Float64(l * l) * Float64(U - U_42_)) / Om)))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.25e-163)
		tmp = sqrt((n * (t * (2.0 * U))));
	elseif (l <= 7.2e-22)
		tmp = ((2.0 * (n * U)) * (t - ((n / Om) * (((l * l) * (U - U_42_)) / Om)))) ^ 0.5;
	else
		tmp = (2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.25e-163], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.2e-22], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(n / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.2499999999999999e-163

   1. Initial program 50.4%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in t around inf 40.8%

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

      1. associate-*r* 39.0%

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

   6. Simplified 39.0%

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

      1. associate-*l* 40.8%

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

      2. *-commutative 40.8%

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

      3. expm1-log1p-u 39.7%

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

      4. expm1-udef 22.2%

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

   8. Applied egg-rr 22.2%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 40.8%

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

      3. *-commutative 40.8%

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

      4. associate-*l* 40.9%

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

      5. *-commutative 40.9%

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

      6. associate-*r* 40.9%

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

   10. Simplified 40.9%

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

   if 2.2499999999999999e-163 < l < 7.1999999999999996e-22

   1. Initial program 61.1%

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

   3. Simplified 53.1%

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

      1. pow1/2 53.1%

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

      2. fma-udef 53.1%

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

      3. associate-*l/ 53.1%

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

      4. associate-*r* 61.1%

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

      5. *-commutative 61.1%

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

      6. associate--l- 61.1%

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

      7. cancel-sign-sub-inv 61.1%

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

      8. metadata-eval 61.1%

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

      9. associate-*r* 61.1%

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

   5. Applied egg-rr 61.1%

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

   6. Taylor expanded in Om around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. unpow2 63.9%

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

      3. unpow2 63.9%

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

      4. times-frac 61.5%

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

   8. Simplified 61.5%

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

   if 7.1999999999999996e-22 < l

   1. Initial program 27.6%

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

   3. Simplified 38.0%

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

      1. pow1/2 38.0%

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

      2. fma-udef 38.0%

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

      3. associate-*l/ 26.0%

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

      4. associate-*r* 27.6%

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

      5. *-commutative 27.6%

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

      6. associate--l- 27.6%

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

      7. cancel-sign-sub-inv 27.6%

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

      8. metadata-eval 27.6%

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

      9. associate-*r* 27.7%

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

   5. Applied egg-rr 39.6%

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

   6. Taylor expanded in n around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. cancel-sign-sub-inv 43.4%

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

      3. metadata-eval 43.4%

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

      4. *-commutative 43.4%

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

      5. unpow2 43.4%

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

      6. associate-*r/ 50.1%

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

   8. Simplified 50.1%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-22}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 8: 55.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;U \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot t_1\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot t_1\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= U 2.2e-13)
     (pow (* 2.0 (* n (* U (+ t (* -2.0 t_1))))) 0.5)
     (pow (* (* 2.0 (* n U)) (- t (* 2.0 t_1))) 0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (U <= 2.2e-13) {
		tmp = pow((2.0 * (n * (U * (t + (-2.0 * t_1))))), 0.5);
	} else {
		tmp = pow(((2.0 * (n * U)) * (t - (2.0 * t_1))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / om)
    if (u <= 2.2d-13) then
        tmp = (2.0d0 * (n * (u * (t + ((-2.0d0) * t_1))))) ** 0.5d0
    else
        tmp = ((2.0d0 * (n * u)) * (t - (2.0d0 * t_1))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (U <= 2.2e-13) {
		tmp = Math.pow((2.0 * (n * (U * (t + (-2.0 * t_1))))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * (n * U)) * (t - (2.0 * t_1))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if U <= 2.2e-13:
		tmp = math.pow((2.0 * (n * (U * (t + (-2.0 * t_1))))), 0.5)
	else:
		tmp = math.pow(((2.0 * (n * U)) * (t - (2.0 * t_1))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (U <= 2.2e-13)
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * t_1))))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(2.0 * t_1))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (U <= 2.2e-13)
		tmp = (2.0 * (n * (U * (t + (-2.0 * t_1))))) ^ 0.5;
	else
		tmp = ((2.0 * (n * U)) * (t - (2.0 * t_1))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, 2.2e-13], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 2.19999999999999997e-13

   1. Initial program 44.2%

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

   3. Simplified 44.6%

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

      1. pow1/2 44.6%

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

      2. fma-udef 44.6%

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

      3. associate-*l/ 41.7%

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

      4. associate-*r* 44.2%

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

      5. *-commutative 44.2%

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

      6. associate--l- 44.2%

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

      7. cancel-sign-sub-inv 44.2%

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

      8. metadata-eval 44.2%

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

      9. associate-*r* 43.7%

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

   5. Applied egg-rr 47.0%

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

   6. Taylor expanded in n around 0 45.5%

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

      1. *-commutative 45.5%

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

      2. cancel-sign-sub-inv 45.5%

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

      3. metadata-eval 45.5%

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

      4. *-commutative 45.5%

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

      5. unpow2 45.5%

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

      6. associate-*r/ 49.7%

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

   8. Simplified 49.7%

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

   if 2.19999999999999997e-13 < U

   1. Initial program 57.0%

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

   3. Simplified 64.8%

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

      1. pow1/2 64.8%

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

      2. fma-udef 64.8%

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

      3. associate-*l/ 54.9%

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

      4. associate-*r* 57.1%

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

      5. *-commutative 57.1%

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

      6. associate--l- 57.1%

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

      7. cancel-sign-sub-inv 57.1%

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

      8. metadata-eval 57.1%

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

      9. associate-*r* 57.1%

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

   5. Applied egg-rr 67.0%

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

   6. Taylor expanded in Om around inf 57.5%

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

      1. unpow2 57.5%

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

      2. associate-*r/ 65.4%

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

   8. Simplified 65.4%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 9: 42.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-107}:\\ \;\;\;\;{\left(n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.2e-107)
   (pow (* n (* t (* 2.0 U))) 0.5)
   (sqrt (* (* 2.0 (* n U)) (- t (/ 2.0 (/ (/ Om l) l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.2e-107) {
		tmp = pow((n * (t * (2.0 * U))), 0.5);
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 / ((Om / 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 <= 2.2d-107) then
        tmp = (n * (t * (2.0d0 * u))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (2.0d0 / ((om / 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 <= 2.2e-107) {
		tmp = Math.pow((n * (t * (2.0 * U))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - (2.0 / ((Om / l) / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.2e-107:
		tmp = math.pow((n * (t * (2.0 * U))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - (2.0 / ((Om / l) / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.2e-107)
		tmp = Float64(n * Float64(t * Float64(2.0 * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(2.0 / Float64(Float64(Om / l) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.2e-107)
		tmp = (n * (t * (2.0 * U))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 / ((Om / l) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.2e-107], N[Power[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 / N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.20000000000000012e-107

   1. Initial program 52.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around inf 40.9%

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

      1. associate-*r* 40.3%

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

   6. Simplified 40.3%

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

      1. associate-*l* 40.9%

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

      2. *-commutative 40.9%

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

      3. expm1-log1p-u 39.7%

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

      4. expm1-udef 22.6%

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

   8. Applied egg-rr 22.6%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 40.9%

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

      3. *-commutative 40.9%

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

      4. associate-*l* 41.0%

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

      5. *-commutative 41.0%

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

      6. associate-*r* 41.0%

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

   10. Simplified 41.0%

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

       1. pow1/2 42.1%

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

       2. *-commutative 42.1%

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

       3. *-commutative 42.1%

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

   12. Applied egg-rr 42.1%

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

   if 2.20000000000000012e-107 < l

   1. Initial program 34.3%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in Om around inf 29.2%

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

      1. associate-*r/ 29.2%

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

      2. associate-/l* 29.2%

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

      3. unpow2 29.2%

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

      4. associate-/r* 35.7%

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

   6. Simplified 35.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-107}:\\ \;\;\;\;{\left(n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2}{\frac{\frac{Om}{\ell}}{\ell}}\right)}\\ \end{array} \]

Alternative 10: 53.9% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (pow (* 2.0 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))) 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 + (-2.0 * (l * (l / Om))))))), 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 + ((-2.0d0) * (l * (l / om))))))) ** 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 + (-2.0 * (l * (l / Om))))))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.7%

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

3. Simplified 48.4%

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

   1. pow1/2 48.4%

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

   2. fma-udef 48.4%

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

   3. associate-*l/ 44.3%

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

   4. associate-*r* 46.7%

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

   5. *-commutative 46.7%

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

   6. associate--l- 46.7%

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

   7. cancel-sign-sub-inv 46.7%

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

   8. metadata-eval 46.7%

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

   9. associate-*r* 46.3%

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

5. Applied egg-rr 50.8%

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

6. Taylor expanded in n around 0 46.6%

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

   1. *-commutative 46.6%

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

   2. cancel-sign-sub-inv 46.6%

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

   3. metadata-eval 46.6%

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

   4. *-commutative 46.6%

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

   5. unpow2 46.6%

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

   6. associate-*r/ 50.4%

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

8. Simplified 50.4%

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

9. Final simplification 50.4%

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

Alternative 11: 37.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-110}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e-110)
   (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 (l <= 1e-110) {
		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 (l <= 1d-110) 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 (l <= 1e-110) {
		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 l <= 1e-110:
		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 (l <= 1e-110)
		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 (l <= 1e-110)
		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[l, 1e-110], 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 l < 1.0000000000000001e-110

   1. Initial program 52.1%

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

   3. Simplified 51.2%

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

      1. pow1/2 51.2%

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

      2. fma-udef 51.2%

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

      3. associate-*l/ 49.1%

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

      4. associate-*r* 52.1%

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

      5. *-commutative 52.1%

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

      6. associate--l- 52.1%

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

      7. cancel-sign-sub-inv 52.1%

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

      8. metadata-eval 52.1%

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

      9. associate-*r* 51.5%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in t around inf 42.2%

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

   if 1.0000000000000001e-110 < l

   1. Initial program 35.1%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in t around inf 18.9%

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

      1. associate-*r* 23.5%

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

   6. Simplified 23.5%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-110}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 12: 37.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-110}:\\ \;\;\;\;{\left(n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e-110)
   (pow (* n (* t (* 2.0 U))) 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 (l <= 2e-110) {
		tmp = pow((n * (t * (2.0 * U))), 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 (l <= 2d-110) then
        tmp = (n * (t * (2.0d0 * u))) ** 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 (l <= 2e-110) {
		tmp = Math.pow((n * (t * (2.0 * U))), 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 l <= 2e-110:
		tmp = math.pow((n * (t * (2.0 * U))), 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 (l <= 2e-110)
		tmp = Float64(n * Float64(t * Float64(2.0 * U))) ^ 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 (l <= 2e-110)
		tmp = (n * (t * (2.0 * U))) ^ 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[l, 2e-110], N[Power[N[(n * N[(t * N[(2.0 * U), $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 l < 2.0000000000000001e-110

   1. Initial program 52.1%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in t around inf 41.1%

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

      1. associate-*r* 39.9%

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

   6. Simplified 39.9%

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

      1. associate-*l* 41.1%

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

      2. *-commutative 41.1%

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

      3. expm1-log1p-u 39.9%

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

      4. expm1-udef 22.7%

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

   8. Applied egg-rr 22.7%

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

      1. expm1-def 39.9%

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

      2. expm1-log1p 41.1%

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

      3. *-commutative 41.1%

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

      4. associate-*l* 41.2%

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

      5. *-commutative 41.2%

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

      6. associate-*r* 41.2%

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

   10. Simplified 41.2%

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

       1. pow1/2 42.3%

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

       2. *-commutative 42.3%

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

       3. *-commutative 42.3%

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

   12. Applied egg-rr 42.3%

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

   if 2.0000000000000001e-110 < l

   1. Initial program 35.1%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in t around inf 18.9%

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

      1. associate-*r* 23.5%

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

   6. Simplified 23.5%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-110}:\\ \;\;\;\;{\left(n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 13: 36.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.4 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.4e-116)
   (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 (l <= 7.4e-116) {
		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 (l <= 7.4d-116) 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 (l <= 7.4e-116) {
		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 l <= 7.4e-116:
		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 (l <= 7.4e-116)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 7.4e-116)
		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[l, 7.4e-116], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.4000000000000005e-116

   1. Initial program 52.1%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in t around inf 41.0%

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

   if 7.4000000000000005e-116 < l

   1. Initial program 35.5%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in t around inf 19.6%

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

      1. associate-*r* 24.1%

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

   6. Simplified 24.1%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.4 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 14: 36.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.5e-116)
   (sqrt (* n (* t (* 2.0 U))))
   (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 (l <= 7.5e-116) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} 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 (l <= 7.5d-116) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    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 (l <= 7.5e-116) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7.5e-116:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	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 (l <= 7.5e-116)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	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 (l <= 7.5e-116)
		tmp = sqrt((n * (t * (2.0 * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.5e-116], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.5000000000000004e-116

   1. Initial program 52.1%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in t around inf 41.0%

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

      1. associate-*r* 39.8%

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

   6. Simplified 39.8%

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

      1. associate-*l* 41.0%

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

      2. *-commutative 41.0%

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

      3. expm1-log1p-u 39.8%

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

      4. expm1-udef 23.0%

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

   8. Applied egg-rr 23.0%

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

      1. expm1-def 39.8%

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

      2. expm1-log1p 41.0%

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

      3. *-commutative 41.0%

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

      4. associate-*l* 41.1%

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

      5. *-commutative 41.1%

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

      6. associate-*r* 41.1%

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

   10. Simplified 41.1%

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

   if 7.5000000000000004e-116 < l

   1. Initial program 35.5%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in t around inf 19.6%

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

      1. associate-*r* 24.1%

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

   6. Simplified 24.1%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 15: 35.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.7%

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

3. Simplified 48.4%

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

4. Taylor expanded in t around inf 34.0%

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

5. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2023274 
(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*))))))