Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 61.4%

Time: 20.8s

Alternatives: 11

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: 49.6% 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: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot 2\right) \cdot U\\ t_4 := t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_4 \leq -5 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot {\left(\frac{\ell}{Om} \cdot \sqrt{U - U*}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot t_1\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, t_1, t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) Om))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3 (* (* n 2.0) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_2 (- U* U))))))
   (if (<= t_4 -5e+120)
     (sqrt
      (*
       (* n 2.0)
       (*
        U
        (-
         (+ t (* -2.0 (/ l (/ Om l))))
         (* n (pow (* (/ l Om) (sqrt (- U U*))) 2.0))))))
     (if (<= t_4 0.0)
       (* (sqrt (* n 2.0)) (sqrt (* U (- t (* 2.0 t_1)))))
       (if (<= t_4 INFINITY)
         (sqrt (* t_3 (- (- t (* 2.0 (* l (/ l Om)))) (* (- U U*) t_2))))
         (pow (* 2.0 (* (* n U) (fma -2.0 t_1 t))) 0.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow(l, 2.0) / Om;
	double t_2 = n * pow((l / Om), 2.0);
	double t_3 = (n * 2.0) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)));
	double tmp;
	if (t_4 <= -5e+120) {
		tmp = sqrt(((n * 2.0) * (U * ((t + (-2.0 * (l / (Om / l)))) - (n * pow(((l / Om) * sqrt((U - U_42_))), 2.0))))));
	} else if (t_4 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - (2.0 * t_1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * t_2))));
	} else {
		tmp = pow((2.0 * ((n * U) * fma(-2.0, t_1, t))), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64((l ^ 2.0) / Om)
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = Float64(Float64(n * 2.0) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_2 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_4 <= -5e+120)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) - Float64(n * (Float64(Float64(l / Om) * sqrt(Float64(U - U_42_))) ^ 2.0))))));
	elseif (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - Float64(2.0 * t_1)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(Float64(U - U_42_) * t_2))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * fma(-2.0, t_1, t))) ^ 0.5;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+120], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(n * N[Power[N[(N[(l / Om), $MachinePrecision] * N[Sqrt[N[(U - U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < -5.00000000000000019e120

   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 66.4%

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

      1. add-sqr-sqrt 66.4%

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

      2. pow2 66.4%

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

      3. sqrt-prod 66.1%

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

      4. unpow2 66.1%

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

      5. sqrt-prod 49.7%

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

      6. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -5.00000000000000019e120 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 0.0

   1. Initial program 8.0%

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

   3. Simplified 8.0%

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

   5. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in n around 0 48.2%

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

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

   1. Initial program 67.9%

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

      1. associate-*l/ 74.5%

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

   3. Applied egg-rr 74.5%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in n around 0 2.1%

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

      1. pow1/2 20.7%

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

      2. associate-*r* 19.7%

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

      3. cancel-sign-sub-inv 19.7%

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

      4. metadata-eval 19.7%

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

      5. unpow2 19.7%

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

      6. associate-*l/ 19.6%

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

      7. associate-/r/ 19.6%

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

      8. +-commutative 19.6%

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

      9. associate-/r/ 19.6%

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

      10. associate-*l/ 19.7%

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

      11. unpow2 19.7%

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

      12. fma-def 19.7%

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

   6. Applied egg-rr 19.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq -5 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot {\left(\frac{\ell}{Om} \cdot \sqrt{U - U*}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 2: 60.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.0%

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

   3. Simplified 8.0%

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

   5. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in l around 0 45.8%

      \[\leadsto \sqrt{n \cdot 2} \cdot \color{blue}{\sqrt{U \cdot t}} \]

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

   1. Initial program 67.9%

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

      1. associate-*l/ 74.5%

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

   3. Applied egg-rr 74.5%

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

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

   1. Initial program 0.0%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in n around 0 1.8%

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

      1. pow1/2 20.5%

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

      2. associate-*r* 19.6%

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

      3. cancel-sign-sub-inv 19.6%

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

      4. metadata-eval 19.6%

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

      5. unpow2 19.6%

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

      6. associate-*l/ 23.4%

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

      7. associate-/r/ 23.4%

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

      8. +-commutative 23.4%

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

      9. associate-/r/ 23.4%

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

      10. associate-*l/ 19.6%

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

      11. unpow2 19.6%

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

      12. fma-def 19.6%

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

   6. Applied egg-rr 19.6%

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

4. Final simplification 60.3%

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

Alternative 3: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ t_2 := \left(n \cdot 2\right) \cdot U\\ t_3 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_3 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - 2 \cdot t_1\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(-2, t_1, t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) Om))
        (t_2 (* (* n 2.0) U))
        (t_3 (* n (pow (/ l Om) 2.0)))
        (t_4 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_3 (- U* U)))))))
   (if (<= t_4 0.0)
     (* (sqrt (* n 2.0)) (sqrt (* U (- t (* 2.0 t_1)))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_2 (- (- t (* 2.0 (* l (/ l Om)))) (* (- U U*) t_3))))
       (pow (* 2.0 (* (* n U) (fma -2.0 t_1 t))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow(l, 2.0) / Om;
	double t_2 = (n * 2.0) * U;
	double t_3 = n * pow((l / Om), 2.0);
	double t_4 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)))));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - (2.0 * t_1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * t_3))));
	} else {
		tmp = pow((2.0 * ((n * U) * fma(-2.0, t_1, t))), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64((l ^ 2.0) / Om)
	t_2 = Float64(Float64(n * 2.0) * U)
	t_3 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_4 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_3 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - Float64(2.0 * t_1)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(Float64(U - U_42_) * t_3))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * fma(-2.0, t_1, t))) ^ 0.5;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.0%

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

   3. Simplified 8.0%

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

   5. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in n around 0 48.2%

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

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

   1. Initial program 67.9%

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

      1. associate-*l/ 74.5%

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

   3. Applied egg-rr 74.5%

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

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

   1. Initial program 0.0%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in n around 0 1.8%

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

      1. pow1/2 20.5%

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

      2. associate-*r* 19.6%

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

      3. cancel-sign-sub-inv 19.6%

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

      4. metadata-eval 19.6%

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

      5. unpow2 19.6%

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

      6. associate-*l/ 23.4%

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

      7. associate-/r/ 23.4%

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

      8. +-commutative 23.4%

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

      9. associate-/r/ 23.4%

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

      10. associate-*l/ 19.6%

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

      11. unpow2 19.6%

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

      12. fma-def 19.6%

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

   6. Applied egg-rr 19.6%

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

4. Final simplification 60.6%

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

Alternative 4: 55.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = (n * 2.0) * U;
	double tmp;
	if (sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U))))) <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	} else {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * 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) :: t_2
    real(8) :: tmp
    t_1 = n * ((l / om) ** 2.0d0)
    t_2 = (n * 2.0d0) * u
    if (sqrt((t_2 * ((t - (2.0d0 * ((l * l) / om))) + (t_1 * (u_42 - u))))) <= 0.0d0) then
        tmp = sqrt((n * 2.0d0)) * sqrt((u * t))
    else
        tmp = sqrt((t_2 * ((t - (2.0d0 * (l * (l / om)))) - ((u - u_42) * 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 = n * Math.pow((l / Om), 2.0);
	double t_2 = (n * 2.0) * U;
	double tmp;
	if (Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U))))) <= 0.0) {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((U * t));
	} else {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - ((U - U_42_) * t_1))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.0%

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

   3. Simplified 8.0%

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

   5. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in l around 0 45.8%

      \[\leadsto \sqrt{n \cdot 2} \cdot \color{blue}{\sqrt{U \cdot t}} \]

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

   1. Initial program 53.1%

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

      1. associate-*l/ 60.2%

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

   3. Applied egg-rr 60.2%

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

4. Final simplification 58.2%

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

Alternative 5: 56.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;n \leq -2.45 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* (pow (/ l Om) 2.0) (- U U*)))))
   (if (<= n -2.45e-303)
     (sqrt (* (* n 2.0) (* U (- (+ t (* -2.0 (/ l (/ Om l)))) t_1))))
     (* (sqrt (* n 2.0)) (sqrt (* U (- t (fma 2.0 (* l (/ l Om)) t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (pow((l / Om), 2.0) * (U - U_42_));
	double tmp;
	if (n <= -2.45e-303) {
		tmp = sqrt(((n * 2.0) * (U * ((t + (-2.0 * (l / (Om / l)))) - t_1))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t - fma(2.0, (l * (l / Om)), t_1))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_)))
	tmp = 0.0
	if (n <= -2.45e-303)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) - t_1))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), t_1)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.45e-303], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.45e-303

   1. Initial program 48.9%

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

   3. Simplified 56.4%

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

   if -2.45e-303 < n

   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 49.3%

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

   5. Applied egg-rr 60.1%

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

      1. *-commutative 60.1%

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

      2. *-commutative 60.1%

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

   7. Simplified 60.1%

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

      1. unpow2 44.0%

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

      2. associate-*l/ 49.3%

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

   9. Applied egg-rr 64.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.45 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 1.8e+112)
   (sqrt
    (*
     (* n 2.0)
     (*
      U
      (-
       (+ t (* -2.0 (/ l (/ Om l))))
       (* n (* (pow (/ l Om) 2.0) (- U U*)))))))
   (* (sqrt (* 2.0 U)) (sqrt (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.8e+112) {
		tmp = sqrt(((n * 2.0) * (U * ((t + (-2.0 * (l / (Om / l)))) - (n * (pow((l / Om), 2.0) * (U - U_42_)))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((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 (u <= 1.8d+112) then
        tmp = sqrt(((n * 2.0d0) * (u * ((t + ((-2.0d0) * (l / (om / l)))) - (n * (((l / om) ** 2.0d0) * (u - u_42)))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((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 (U <= 1.8e+112) {
		tmp = Math.sqrt(((n * 2.0) * (U * ((t + (-2.0 * (l / (Om / l)))) - (n * (Math.pow((l / Om), 2.0) * (U - U_42_)))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 1.8e+112:
		tmp = math.sqrt(((n * 2.0) * (U * ((t + (-2.0 * (l / (Om / l)))) - (n * (math.pow((l / Om), 2.0) * (U - U_42_)))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 1.8e+112)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) - Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.8e+112)
		tmp = sqrt(((n * 2.0) * (U * ((t + (-2.0 * (l / (Om / l)))) - (n * (((l / Om) ^ 2.0) * (U - U_42_)))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 1.8e+112], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.8e112

   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 55.1%

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

   if 1.8e112 < U

   1. Initial program 53.8%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in t around inf 50.3%

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

      1. pow1/2 53.5%

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

      2. associate-*r* 53.5%

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

      3. unpow-prod-down 68.5%

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

      4. pow1/2 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. unpow1/2 68.5%

         \[\leadsto \color{blue}{\sqrt{2 \cdot U}} \cdot \sqrt{n \cdot t} \]

   8. Simplified 68.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 7: 47.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{+267}:\\ \;\;\;\;{\left({\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot {\left(\frac{\frac{Om}{\ell}}{\ell}\right)}^{-1}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -7.8e+267)
   (pow (pow (* t (* n (* 2.0 U))) 3.0) 0.16666666666666666)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (pow (/ (/ Om l) l) -1.0)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -7.8e+267) {
		tmp = pow(pow((t * (n * (2.0 * U))), 3.0), 0.16666666666666666);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * pow(((Om / l) / l), -1.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-7.8d+267)) then
        tmp = ((t * (n * (2.0d0 * u))) ** 3.0d0) ** 0.16666666666666666d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (((om / l) / l) ** (-1.0d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -7.8e+267) {
		tmp = Math.pow(Math.pow((t * (n * (2.0 * U))), 3.0), 0.16666666666666666);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * Math.pow(((Om / l) / l), -1.0)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -7.8e+267:
		tmp = math.pow(math.pow((t * (n * (2.0 * U))), 3.0), 0.16666666666666666)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * math.pow(((Om / l) / l), -1.0)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -7.8e+267)
		tmp = (Float64(t * Float64(n * Float64(2.0 * U))) ^ 3.0) ^ 0.16666666666666666;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * (Float64(Float64(Om / l) / l) ^ -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -7.8e+267)
		tmp = ((t * (n * (2.0 * U))) ^ 3.0) ^ 0.16666666666666666;
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (((Om / l) / l) ^ -1.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -7.8e+267], N[Power[N[Power[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[Power[N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.79999999999999961e267

   1. Initial program 51.6%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in t around inf 18.2%

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

      1. add-cbrt-cube 18.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. pow1/3 18.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 18.2%

         \[\leadsto {\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333} \]

      4. pow1 18.2%

         \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 51.6%

         \[\leadsto {\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 51.6%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-*r* 51.6%

         \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 51.6%

         \[\leadsto {\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 51.6%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 51.6%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{1.5}}} \]

      2. *-commutative 51.6%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}}^{1.5}} \]

   8. Simplified 51.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{1.5}}} \]
   9. Step-by-step derivation

      1. pow1/3 51.6%

         \[\leadsto \color{blue}{{\left({\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. *-commutative 51.6%

         \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}}^{1.5}\right)}^{0.3333333333333333} \]

      3. pow-pow 51.6%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      4. metadata-eval 51.6%

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

      5. metadata-eval 51.6%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(3 \cdot 0.16666666666666666\right)}} \]

      6. metadata-eval 51.6%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(3 \cdot \color{blue}{\frac{0.3333333333333333}{2}}\right)} \]

      7. pow-pow 68.2%

         \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      8. associate-*r* 68.4%

         \[\leadsto {\left({\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      9. metadata-eval 68.4%

         \[\leadsto {\left({\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   10. Applied egg-rr 68.4%

       \[\leadsto \color{blue}{{\left({\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}^{3}\right)}^{0.16666666666666666}} \]

   if -7.79999999999999961e267 < n

   1. Initial program 46.9%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in n around 0 45.7%

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

      1. unpow2 45.7%

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

      2. associate-*l/ 50.6%

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

      3. associate-/r/ 50.6%

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

      4. clear-num 50.6%

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

      5. inv-pow 50.6%

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

   6. Applied egg-rr 50.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{+267}:\\ \;\;\;\;{\left({\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot {\left(\frac{\frac{Om}{\ell}}{\ell}\right)}^{-1}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+271}:\\ \;\;\;\;{\left({\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.2e+271)
   (pow (pow (* t (* n (* 2.0 U))) 3.0) 0.16666666666666666)
   (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 (n <= -2.2e+271) {
		tmp = pow(pow((t * (n * (2.0 * U))), 3.0), 0.16666666666666666);
	} 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 (n <= (-2.2d+271)) then
        tmp = ((t * (n * (2.0d0 * u))) ** 3.0d0) ** 0.16666666666666666d0
    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 (n <= -2.2e+271) {
		tmp = Math.pow(Math.pow((t * (n * (2.0 * U))), 3.0), 0.16666666666666666);
	} 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 n <= -2.2e+271:
		tmp = math.pow(math.pow((t * (n * (2.0 * U))), 3.0), 0.16666666666666666)
	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 (n <= -2.2e+271)
		tmp = (Float64(t * Float64(n * Float64(2.0 * U))) ^ 3.0) ^ 0.16666666666666666;
	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 (n <= -2.2e+271)
		tmp = ((t * (n * (2.0 * U))) ^ 3.0) ^ 0.16666666666666666;
	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[n, -2.2e+271], N[Power[N[Power[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $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 n < -2.20000000000000001e271

   1. Initial program 51.6%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in t around inf 18.2%

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

      1. add-cbrt-cube 18.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. pow1/3 18.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right) \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 18.2%

         \[\leadsto {\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333} \]

      4. pow1 18.2%

         \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 51.6%

         \[\leadsto {\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 51.6%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-*r* 51.6%

         \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 51.6%

         \[\leadsto {\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 51.6%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 51.6%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{1.5}}} \]

      2. *-commutative 51.6%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}}^{1.5}} \]

   8. Simplified 51.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{1.5}}} \]
   9. Step-by-step derivation

      1. pow1/3 51.6%

         \[\leadsto \color{blue}{{\left({\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. *-commutative 51.6%

         \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}}^{1.5}\right)}^{0.3333333333333333} \]

      3. pow-pow 51.6%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      4. metadata-eval 51.6%

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

      5. metadata-eval 51.6%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\left(3 \cdot 0.16666666666666666\right)}} \]

      6. metadata-eval 51.6%

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(3 \cdot \color{blue}{\frac{0.3333333333333333}{2}}\right)} \]

      7. pow-pow 68.2%

         \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      8. associate-*r* 68.4%

         \[\leadsto {\left({\color{blue}{\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      9. metadata-eval 68.4%

         \[\leadsto {\left({\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   10. Applied egg-rr 68.4%

       \[\leadsto \color{blue}{{\left({\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}^{3}\right)}^{0.16666666666666666}} \]

   if -2.20000000000000001e271 < n

   1. Initial program 46.9%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in n around 0 45.7%

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

      1. unpow2 45.7%

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

      2. associate-*l/ 50.6%

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

   6. Applied egg-rr 50.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+271}:\\ \;\;\;\;{\left({\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 9: 47.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (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_) {
	return sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
}

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := 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. Initial program 47.0%

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

3. Simplified 51.5%

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

4. Taylor expanded in n around 0 44.7%

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

   1. unpow2 44.7%

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

   2. associate-*l/ 49.4%

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

6. Applied egg-rr 49.4%

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

7. Final simplification 49.4%

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

Alternative 10: 37.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.0%

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

3. Simplified 51.5%

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

4. Taylor expanded in t around inf 35.6%

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

   1. pow1/2 37.2%

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

   2. associate-*r* 37.2%

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

6. Applied egg-rr 37.2%

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

7. Final simplification 37.2%

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

Alternative 11: 35.7% 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 47.0%

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

3. Simplified 51.5%

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

4. Taylor expanded in t around inf 35.6%

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

5. Final simplification 35.6%

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

Reproduce

herbie shell --seed 2023305 
(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*))))))