Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 60.0%

Time: 24.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.3% 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: 60.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, t\_1 \cdot \left(n \cdot U\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(t\_1 \cdot U*\right) \cdot \left(-n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 11.3%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in U around inf 26.7%

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

      1. associate-/l* 26.7%

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

      2. unpow2 26.7%

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

      3. unpow2 26.7%

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

      4. times-frac 32.3%

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

      5. unpow2 32.3%

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

   7. Simplified 32.3%

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

      1. pow1/2 32.3%

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

      2. *-commutative 32.3%

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

      3. unpow-prod-down 40.5%

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

      4. pow1/2 40.5%

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

      5. associate-*r/ 40.5%

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

      6. unpow2 40.5%

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

      7. associate-*r* 40.5%

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

      8. pow1/2 40.5%

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

   9. Applied egg-rr 40.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 69.6%

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

   3. Simplified 76.9%

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

      1. associate-*r* 76.9%

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

      2. fma-define 76.9%

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

      3. associate-*r* 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in U around 0 59.2%

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

      1. associate-/l* 62.7%

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

      2. unpow2 62.7%

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

      3. unpow2 62.7%

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

      4. times-frac 77.4%

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

      5. unpow2 77.4%

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

      6. neg-mul-1 77.4%

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

      7. distribute-lft-neg-out 77.4%

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

      8. *-commutative 77.4%

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

   9. Simplified 77.4%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 0.0%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in Om around -inf 14.8%

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

      1. +-commutative 14.8%

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

      2. mul-1-neg 14.8%

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

      3. unsub-neg 14.8%

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

      4. fma-define 14.8%

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

      5. associate-/l* 16.8%

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

      6. associate-/l* 16.8%

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

   7. Simplified 16.8%

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

   8. Taylor expanded in U* around inf 28.9%

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

      1. mul-1-neg 28.9%

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

      2. associate-/l* 28.9%

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

      3. associate-/l* 28.8%

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

      4. associate-/l* 30.7%

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

   10. Simplified 30.7%

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

4. Final simplification 63.3%

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

Alternative 2: 60.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(t\_1 \cdot U*\right) \cdot \left(-n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in n around 0 28.8%

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

      1. unpow2 28.8%

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

      2. associate-*r/ 33.3%

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

      3. *-commutative 33.3%

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

   7. Applied egg-rr 33.3%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 69.6%

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

   3. Simplified 76.9%

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

      1. associate-*r* 76.9%

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

      2. fma-define 76.9%

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

      3. associate-*r* 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in U around 0 59.2%

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

      1. associate-/l* 62.7%

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

      2. unpow2 62.7%

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

      3. unpow2 62.7%

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

      4. times-frac 77.4%

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

      5. unpow2 77.4%

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

      6. neg-mul-1 77.4%

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

      7. distribute-lft-neg-out 77.4%

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

      8. *-commutative 77.4%

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

   9. Simplified 77.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in Om around -inf 15.2%

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

      1. +-commutative 15.2%

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

      2. mul-1-neg 15.2%

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

      3. unsub-neg 15.2%

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

      4. fma-define 15.2%

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

      5. associate-/l* 17.6%

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

      6. associate-/l* 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in U* around inf 34.0%

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

      1. mul-1-neg 34.0%

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

      2. associate-/l* 33.9%

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

      3. associate-/l* 33.8%

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

      4. associate-/l* 36.1%

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

   10. Simplified 36.1%

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

4. Final simplification 63.2%

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

Alternative 3: 60.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left(t\_1 \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in n around 0 28.8%

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

      1. unpow2 28.8%

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

      2. associate-*r/ 33.3%

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

      3. *-commutative 33.3%

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

   7. Applied egg-rr 33.3%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 69.6%

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

   3. Simplified 76.9%

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

      1. associate-*r* 76.9%

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

      2. fma-define 76.9%

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

      3. associate-*r* 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in U around 0 59.2%

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

      1. associate-/l* 62.7%

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

      2. unpow2 62.7%

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

      3. unpow2 62.7%

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

      4. times-frac 77.4%

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

      5. unpow2 77.4%

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

      6. neg-mul-1 77.4%

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

      7. distribute-lft-neg-out 77.4%

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

      8. *-commutative 77.4%

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

   9. Simplified 77.4%

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

       1. fma-undefine 77.4%

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

   11. Applied egg-rr 77.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in Om around -inf 15.2%

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

      1. +-commutative 15.2%

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

      2. mul-1-neg 15.2%

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

      3. unsub-neg 15.2%

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

      4. fma-define 15.2%

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

      5. associate-/l* 17.6%

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

      6. associate-/l* 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in U* around inf 34.0%

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

      1. mul-1-neg 34.0%

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

      2. associate-/l* 33.9%

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

      3. associate-/l* 33.8%

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

      4. associate-/l* 36.1%

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

   10. Simplified 36.1%

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

4. Final simplification 63.2%

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

Alternative 4: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;Om \leq -8.2 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 2.22 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.3 \cdot 10^{+150}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))))
   (if (<= Om -8.2e+15)
     t_1
     (if (<= Om 2.22e-97)
       (sqrt
        (* (* 2.0 n) (+ (* U t) (/ (* U (* U* (* (pow l 2.0) (/ n Om)))) Om))))
       (if (<= Om 2.3e+150)
         (pow (* 2.0 (* (* n U) (+ t (* (/ (pow l 2.0) Om) -2.0)))) 0.5)
         t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	double tmp;
	if (Om <= -8.2e+15) {
		tmp = t_1;
	} else if (Om <= 2.22e-97) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * (pow(l, 2.0) * (n / Om)))) / Om))));
	} else if (Om <= 2.3e+150) {
		tmp = pow((2.0 * ((n * U) * (t + ((pow(l, 2.0) / Om) * -2.0)))), 0.5);
	} else {
		tmp = t_1;
	}
	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 = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    if (om <= (-8.2d+15)) then
        tmp = t_1
    else if (om <= 2.22d-97) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((u * (u_42 * ((l ** 2.0d0) * (n / om)))) / om))))
    else if (om <= 2.3d+150) then
        tmp = (2.0d0 * ((n * u) * (t + (((l ** 2.0d0) / om) * (-2.0d0))))) ** 0.5d0
    else
        tmp = t_1
    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 = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	double tmp;
	if (Om <= -8.2e+15) {
		tmp = t_1;
	} else if (Om <= 2.22e-97) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * (Math.pow(l, 2.0) * (n / Om)))) / Om))));
	} else if (Om <= 2.3e+150) {
		tmp = Math.pow((2.0 * ((n * U) * (t + ((Math.pow(l, 2.0) / Om) * -2.0)))), 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	tmp = 0
	if Om <= -8.2e+15:
		tmp = t_1
	elif Om <= 2.22e-97:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * (math.pow(l, 2.0) * (n / Om)))) / Om))))
	elif Om <= 2.3e+150:
		tmp = math.pow((2.0 * ((n * U) * (t + ((math.pow(l, 2.0) / Om) * -2.0)))), 0.5)
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))))
	tmp = 0.0
	if (Om <= -8.2e+15)
		tmp = t_1;
	elseif (Om <= 2.22e-97)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(U * Float64(U_42_ * Float64((l ^ 2.0) * Float64(n / Om)))) / Om))));
	elseif (Om <= 2.3e+150)
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0)))) ^ 0.5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	tmp = 0.0;
	if (Om <= -8.2e+15)
		tmp = t_1;
	elseif (Om <= 2.22e-97)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * ((l ^ 2.0) * (n / Om)))) / Om))));
	elseif (Om <= 2.3e+150)
		tmp = (2.0 * ((n * U) * (t + (((l ^ 2.0) / Om) * -2.0)))) ^ 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -8.2e+15], t$95$1, If[LessEqual[Om, 2.22e-97], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(U * N[(U$42$ * N[(N[Power[l, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.3e+150], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -8.2e15 or 2.30000000000000001e150 < Om

   1. Initial program 54.7%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in n around 0 54.1%

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

      1. unpow2 54.1%

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

      2. associate-*r/ 65.7%

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

      3. *-commutative 65.7%

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

   7. Applied egg-rr 65.7%

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

   if -8.2e15 < Om < 2.21999999999999991e-97

   1. Initial program 39.3%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in Om around -inf 45.6%

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

      1. +-commutative 45.6%

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

      2. mul-1-neg 45.6%

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

      3. unsub-neg 45.6%

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

      4. fma-define 45.6%

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

      5. associate-/l* 49.8%

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

      6. associate-/l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in U* around inf 51.2%

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

      1. mul-1-neg 51.2%

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

      2. associate-/l* 54.1%

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

      3. associate-/l* 54.0%

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

      4. associate-/l* 52.0%

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

   10. Simplified 52.0%

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

   if 2.21999999999999991e-97 < Om < 2.30000000000000001e150

   1. Initial program 45.9%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in n around 0 39.3%

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

      1. pow1/2 47.5%

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

      2. associate-*r* 54.1%

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

      3. *-commutative 54.1%

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

      4. cancel-sign-sub-inv 54.1%

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

      5. metadata-eval 54.1%

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

   7. Applied egg-rr 54.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8.2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.22 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.3 \cdot 10^{+150}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.3 \cdot 10^{-15} \lor \neg \left(Om \leq 6.1 \cdot 10^{+150}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.3e-15) (not (<= Om 6.1e+150)))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
   (pow (* 2.0 (* (* n U) (+ t (* (/ (pow l 2.0) Om) -2.0)))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.3e-15) || !(Om <= 6.1e+150)) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = pow((2.0 * ((n * U) * (t + ((pow(l, 2.0) / Om) * -2.0)))), 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 ((om <= (-1.3d-15)) .or. (.not. (om <= 6.1d+150))) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else
        tmp = (2.0d0 * ((n * u) * (t + (((l ** 2.0d0) / om) * (-2.0d0))))) ** 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 ((Om <= -1.3e-15) || !(Om <= 6.1e+150)) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = Math.pow((2.0 * ((n * U) * (t + ((Math.pow(l, 2.0) / Om) * -2.0)))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.3e-15) or not (Om <= 6.1e+150):
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	else:
		tmp = math.pow((2.0 * ((n * U) * (t + ((math.pow(l, 2.0) / Om) * -2.0)))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.3e-15) || !(Om <= 6.1e+150))
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0)))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.3e-15) || ~((Om <= 6.1e+150)))
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	else
		tmp = (2.0 * ((n * U) * (t + (((l ^ 2.0) / Om) * -2.0)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.3e-15], N[Not[LessEqual[Om, 6.1e+150]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.30000000000000002e-15 or 6.10000000000000026e150 < Om

   1. Initial program 53.9%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in n around 0 54.0%

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

      1. unpow2 54.0%

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

      2. associate-*r/ 65.3%

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

      3. *-commutative 65.3%

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

   7. Applied egg-rr 65.3%

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

   if -1.30000000000000002e-15 < Om < 6.10000000000000026e150

   1. Initial program 42.2%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in n around 0 32.5%

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

      1. pow1/2 44.7%

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

      2. associate-*r* 47.0%

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

      3. *-commutative 47.0%

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

      4. cancel-sign-sub-inv 47.0%

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

      5. metadata-eval 47.0%

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

   7. Applied egg-rr 47.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.3 \cdot 10^{-15} \lor \neg \left(Om \leq 6.1 \cdot 10^{+150}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e-79)
   (sqrt (* 2.0 (fabs (* t (* n U)))))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-79) {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2d-79) then
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-79) {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2e-79:
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2e-79)
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2e-79)
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2e-79], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2e-79

   1. Initial program 52.3%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in t around inf 40.3%

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

      1. associate-*r* 42.0%

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

   7. Simplified 42.0%

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

      1. pow1 42.0%

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

      2. associate-*r* 40.3%

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

      3. add-sqr-sqrt 40.2%

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

      4. unpow2 40.2%

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

      5. metadata-eval 40.2%

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

      6. pow-prod-up 40.2%

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

      7. pow-prod-down 28.4%

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

      8. pow-prod-down 28.4%

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

      9. add-sqr-sqrt 32.4%

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

   9. Applied egg-rr 32.4%

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

       1. unpow1/2 32.4%

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

       2. unpow2 32.4%

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

       3. rem-sqrt-square 42.9%

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

       4. associate-*r* 45.7%

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

   11. Simplified 45.7%

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

   if 2e-79 < l

   1. Initial program 38.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in n around 0 41.2%

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

      1. unpow2 41.2%

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

      2. associate-*r/ 49.9%

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

      3. *-commutative 49.9%

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

   7. Applied egg-rr 49.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-79}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.7e-79)
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.7e-79) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.7d-79) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.7e-79) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.7e-79:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.7e-79)
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.7e-79)
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.7e-79], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7000000000000002e-79

   1. Initial program 52.3%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in t around inf 40.3%

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

      1. associate-*r* 42.0%

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

   7. Simplified 42.0%

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

      1. pow1/2 45.0%

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

      2. associate-*r* 45.0%

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

      3. *-commutative 45.0%

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

      4. associate-*r* 45.0%

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

      5. *-commutative 45.0%

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

   9. Applied egg-rr 45.0%

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

   if 2.7000000000000002e-79 < l

   1. Initial program 38.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in n around 0 41.2%

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

      1. unpow2 41.2%

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

      2. associate-*r/ 49.9%

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

      3. *-commutative 49.9%

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

   7. Applied egg-rr 49.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-79}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-135}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5e-135)
   (pow (* (* 2.0 U) (* n t)) 0.5)
   (pow (* (* (* 2.0 n) U) t) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e-135) {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	} else {
		tmp = pow((((2.0 * n) * U) * t), 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 (t <= (-5d-135)) then
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    else
        tmp = (((2.0d0 * n) * u) * t) ** 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 (t <= -5e-135) {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	} else {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5e-135:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	else:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5e-135)
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	else
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5e-135)
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	else
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5e-135], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0000000000000002e-135

   1. Initial program 49.8%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in t around inf 44.2%

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

      1. pow1/2 46.3%

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

      2. associate-*r* 46.3%

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

   7. Applied egg-rr 46.3%

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

   if -5.0000000000000002e-135 < t

   1. Initial program 47.1%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in t around inf 31.2%

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

      1. associate-*r* 34.2%

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

   7. Simplified 34.2%

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

      1. pow1/2 38.2%

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

      2. associate-*r* 38.2%

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

      3. *-commutative 38.2%

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

      4. associate-*r* 38.2%

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

      5. *-commutative 38.2%

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

   9. Applied egg-rr 38.2%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-135}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-217}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.8e-217)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (pow (* (* 2.0 U) (* n t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.8e-217) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = pow(((2.0 * U) * (n * t)), 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 <= 1.8d-217) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = ((2.0d0 * u) * (n * t)) ** 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 <= 1.8e-217) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.8e-217:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.8e-217)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.8e-217)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.8e-217], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.79999999999999991e-217

   1. Initial program 50.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in t around inf 38.1%

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

      1. add-sqr-sqrt 38.0%

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

      2. pow2 38.0%

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

   7. Applied egg-rr 38.0%

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

      1. unpow2 38.0%

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

      2. add-sqr-sqrt 38.1%

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

      3. *-commutative 38.1%

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

   9. Applied egg-rr 38.1%

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

       1. associate-*l* 38.7%

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

   11. Simplified 38.7%

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

       1. pow1/2 40.8%

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

       2. *-commutative 40.8%

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

   13. Applied egg-rr 40.8%

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

   if 1.79999999999999991e-217 < l

   1. Initial program 44.4%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in t around inf 32.9%

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

      1. pow1/2 34.9%

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

      2. associate-*r* 34.9%

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

   7. Applied egg-rr 34.9%

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

Alternative 10: 36.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{+21}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 2e+21)
   (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 (Om <= 2e+21) {
		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 (om <= 2d+21) 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 (Om <= 2e+21) {
		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 Om <= 2e+21:
		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 (Om <= 2e+21)
		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 (Om <= 2e+21)
		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[Om, 2e+21], 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 Om < 2e21

   1. Initial program 46.0%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in t around inf 29.0%

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

      1. add-sqr-sqrt 29.0%

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

      2. pow2 29.0%

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

   7. Applied egg-rr 29.0%

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

      1. unpow2 29.0%

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

      2. add-sqr-sqrt 29.0%

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

      3. *-commutative 29.0%

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

   9. Applied egg-rr 29.0%

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

       1. associate-*l* 32.5%

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

   11. Simplified 32.5%

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

       1. pow1/2 34.2%

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

       2. *-commutative 34.2%

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

   13. Applied egg-rr 34.2%

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

   if 2e21 < Om

   1. Initial program 52.9%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in t around inf 52.3%

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

Alternative 11: 35.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.4e-240)
   (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 <= 4.4e-240) {
		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 <= 4.4d-240) 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 <= 4.4e-240) {
		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 <= 4.4e-240:
		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 <= 4.4e-240)
		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 <= 4.4e-240)
		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, 4.4e-240], 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 < 4.3999999999999999e-240

   1. Initial program 50.5%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in t around inf 37.7%

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

      1. add-sqr-sqrt 37.7%

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

      2. pow2 37.7%

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

   7. Applied egg-rr 37.7%

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

      1. unpow2 37.7%

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

      2. add-sqr-sqrt 37.7%

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

      3. *-commutative 37.7%

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

   9. Applied egg-rr 37.7%

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

       1. associate-*l* 38.3%

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

   11. Simplified 38.3%

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

   if 4.3999999999999999e-240 < l

   1. Initial program 44.7%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in t around inf 33.7%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.1%

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

3. Simplified 54.7%

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

5. Taylor expanded in t around inf 36.0%

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

Reproduce

herbie shell --seed 2024137 
(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*))))))