Toniolo and Linder, Equation (13)

Percentage Accurate: 49.0% → 57.8%

Time: 17.7s

Alternatives: 17

Speedup: 2.0×

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: 49.0% 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: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.55 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-2 \cdot U\right) \cdot \ell, \left(\frac{\ell}{Om} \cdot n\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right), \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 1.55e+118)
   (sqrt
    (fma
     (* (* -2.0 U) l)
     (* (* (/ l Om) n) (fma (- U U*) (/ n Om) 2.0))
     (* (* (* t n) U) 2.0)))
   (* (sqrt (* (* (fma (* (/ l Om) l) -2.0 t) U) 2.0)) (sqrt n))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1.55e+118) {
		tmp = sqrt(fma(((-2.0 * U) * l), (((l / Om) * n) * fma((U - U_42_), (n / Om), 2.0)), (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) * U) * 2.0)) * sqrt(n);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 1.55e+118)
		tmp = sqrt(fma(Float64(Float64(-2.0 * U) * l), Float64(Float64(Float64(l / Om) * n) * fma(Float64(U - U_42_), Float64(n / Om), 2.0)), Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * U) * 2.0)) * sqrt(n));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 1.55e+118], N[Sqrt[N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.54999999999999993e118

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 61.6%

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

   9. Applied rewrites 67.9%

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

   if 1.54999999999999993e118 < n

   1. Initial program 59.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 61.4

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 66.7%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.55 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-2 \cdot U\right) \cdot \ell, \left(\frac{\ell}{Om} \cdot n\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right), \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot U, \left(\frac{t\_1 \cdot \ell}{Om} \cdot \ell\right) \cdot n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ n Om) (- U U*) 2.0))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          (-
           (* (- U* U) (* (pow (/ l Om) 2.0) n))
           (- (* (/ (* l l) Om) 2.0) t))
          t_2)))
   (if (<= t_3 2e-320)
     (sqrt (fma (* -2.0 U) (* (* (/ (* t_1 l) Om) l) n) (* (* (* t n) U) 2.0)))
     (if (<= t_3 2e+306)
       (sqrt (* (- t (/ (* (* l l) (fma (- U U*) (/ n Om) 2.0)) Om)) t_2))
       (sqrt (* (* (* (* -2.0 U) l) (* (/ l Om) n)) t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((n / Om), (U - U_42_), 2.0);
	double t_2 = (2.0 * n) * U;
	double t_3 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((((l * l) / Om) * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 2e-320) {
		tmp = sqrt(fma((-2.0 * U), ((((t_1 * l) / Om) * l) * n), (((t * n) * U) * 2.0)));
	} else if (t_3 <= 2e+306) {
		tmp = sqrt(((t - (((l * l) * fma((U - U_42_), (n / Om), 2.0)) / Om)) * t_2));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * ((l / Om) * n)) * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(n / Om), Float64(U - U_42_), 2.0)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l * l) / Om) * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 2e-320)
		tmp = sqrt(fma(Float64(-2.0 * U), Float64(Float64(Float64(Float64(t_1 * l) / Om) * l) * n), Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_3 <= 2e+306)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(l * l) * fma(Float64(U - U_42_), Float64(n / Om), 2.0)) / Om)) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(Float64(l / Om) * n)) * t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-320], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision] * n), $MachinePrecision] + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 2e+306], N[Sqrt[N[(N[(t - N[(N[(N[(l * l), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * t$95$1), $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*)))) < 1.99998e-320

   1. Initial program 9.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 53.6%

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

   if 1.99998e-320 < (*.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*)))) < 2.00000000000000003e306

   1. Initial program 96.8%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 92.7%

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

   if 2.00000000000000003e306 < (*.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 24.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 30.1%

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

   10. Applied rewrites 49.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot U, \left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \ell}{Om} \cdot \ell\right) \cdot n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{n}{Om} \cdot \ell\right) \cdot \left(\ell \cdot U\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          (-
           (* (- U* U) (* (pow (/ l Om) 2.0) n))
           (- (* (/ (* l l) Om) 2.0) t))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (fma (* (* (/ n Om) l) (* l U)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_2 2e+306)
       (sqrt (* (- t (/ (* (* l l) (fma (- U U*) (/ n Om) 2.0)) Om)) t_1))
       (sqrt
        (*
         (* (* (* -2.0 U) l) (* (/ l Om) n))
         (fma (/ n Om) (- U U*) 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((((l * l) / Om) * 2.0) - t)) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(fma((((n / Om) * l) * (l * U)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt(((t - (((l * l) * fma((U - U_42_), (n / Om), 2.0)) / Om)) * t_1));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * ((l / Om) * n)) * fma((n / Om), (U - U_42_), 2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l * l) / Om) * 2.0) - t)) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(n / Om) * l) * Float64(l * U)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(l * l) * fma(Float64(U - U_42_), Float64(n / Om), 2.0)) / Om)) * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(Float64(l / Om) * n)) * fma(Float64(n / Om), Float64(U - U_42_), 2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(N[(n / Om), $MachinePrecision] * l), $MachinePrecision] * N[(l * U), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(N[(t - N[(N[(N[(l * l), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $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 8.8%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 36.0

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

   5. Applied rewrites 36.0%

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

   7. Applied rewrites 44.2%

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

   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*)))) < 2.00000000000000003e306

   1. Initial program 96.2%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 92.2%

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

   if 2.00000000000000003e306 < (*.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 24.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 30.1%

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

   10. Applied rewrites 49.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{n}{Om} \cdot \ell\right) \cdot \left(\ell \cdot U\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $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 8.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

   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 70.1%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   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%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 20.1%

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

   7. Applied rewrites 26.8%

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

   9. Applied rewrites 55.8%

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

   10. Taylor expanded in U* around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. unpow2 N/A

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

       10. unswap-sqr N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 26.9

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

   12. Applied rewrites 26.9%

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

4. Final simplification 51.5%

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

Alternative 5: 49.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{U}{Om \cdot Om} \cdot n\right) \cdot \left(\left(\left(U* \cdot n\right) \cdot \ell\right) \cdot \ell\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(N[(N[(U$42$ * n), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 2.0), $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 8.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

   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 70.1%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   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%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 15.4

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

   5. Applied rewrites 15.4%

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

   7. Applied rewrites 16.4%

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

   9. Applied rewrites 19.2%

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

4. Final simplification 50.2%

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

Alternative 6: 37.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot t\_1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          (-
           (* (- U* U) (* (pow (/ l Om) 2.0) n))
           (- (* (/ (* l l) Om) 2.0) t))
          t_1))
        5e-5)
     (sqrt (* (* (* t U) n) 2.0))
     (sqrt (* t_1 t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (sqrt(((((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((((l * l) / Om) * 2.0) - t)) * t_1)) <= 5e-5) {
		tmp = sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = sqrt((t_1 * 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) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (sqrt(((((u_42 - u) * (((l / om) ** 2.0d0) * n)) - ((((l * l) / om) * 2.0d0) - t)) * t_1)) <= 5d-5) then
        tmp = sqrt((((t * u) * n) * 2.0d0))
    else
        tmp = sqrt((t_1 * 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 t_1 = (2.0 * n) * U;
	double tmp;
	if (Math.sqrt(((((U_42_ - U) * (Math.pow((l / Om), 2.0) * n)) - ((((l * l) / Om) * 2.0) - t)) * t_1)) <= 5e-5) {
		tmp = Math.sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = Math.sqrt((t_1 * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if math.sqrt(((((U_42_ - U) * (math.pow((l / Om), 2.0) * n)) - ((((l * l) / Om) * 2.0) - t)) * t_1)) <= 5e-5:
		tmp = math.sqrt((((t * U) * n) * 2.0))
	else:
		tmp = math.sqrt((t_1 * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l * l) / Om) * 2.0) - t)) * t_1)) <= 5e-5)
		tmp = sqrt(Float64(Float64(Float64(t * U) * n) * 2.0));
	else
		tmp = sqrt(Float64(t_1 * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (sqrt(((((U_42_ - U) * (((l / Om) ^ 2.0) * n)) - ((((l * l) / Om) * 2.0) - t)) * t_1)) <= 5e-5)
		tmp = sqrt((((t * U) * n) * 2.0));
	else
		tmp = sqrt((t_1 * t));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 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*))))) < 5.00000000000000024e-5

   1. Initial program 57.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 60.5%

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

   if 5.00000000000000024e-5 < (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 46.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

   7. Applied rewrites 32.6%

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

4. Final simplification 39.9%

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

Alternative 7: 46.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot n}{Om} \cdot \left(\ell \cdot \ell\right), -2, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.8e-135)
   (sqrt (* (* (* t U) n) 2.0))
   (if (<= l 6.8e+67)
     (sqrt
      (*
       (fma
        (* (/ (* (- 2.0 (/ (* U* n) Om)) n) Om) (* l l))
        -2.0
        (* (* t n) 2.0))
       U))
     (sqrt
      (* (* (* (* -2.0 U) l) (fma (/ n Om) (- U U*) 2.0)) (* (/ l Om) n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.8e-135) {
		tmp = sqrt((((t * U) * n) * 2.0));
	} else if (l <= 6.8e+67) {
		tmp = sqrt((fma(((((2.0 - ((U_42_ * n) / Om)) * n) / Om) * (l * l)), -2.0, ((t * n) * 2.0)) * U));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * fma((n / Om), (U - U_42_), 2.0)) * ((l / Om) * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.8e-135)
		tmp = sqrt(Float64(Float64(Float64(t * U) * n) * 2.0));
	elseif (l <= 6.8e+67)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(Float64(2.0 - Float64(Float64(U_42_ * n) / Om)) * n) / Om) * Float64(l * l)), -2.0, Float64(Float64(t * n) * 2.0)) * U));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * fma(Float64(n / Om), Float64(U - U_42_), 2.0)) * Float64(Float64(l / Om) * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.8e-135], N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.8e+67], N[Sqrt[N[(N[(N[(N[(N[(N[(2.0 - N[(N[(U$42$ * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.80000000000000023e-135

   1. Initial program 54.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   7. Applied rewrites 46.5%

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

   if 2.80000000000000023e-135 < l < 6.8000000000000003e67

   1. Initial program 44.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 61.2%

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

   9. Applied rewrites 58.8%

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

   10. Taylor expanded in U around 0

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

   12. Applied rewrites 56.5%

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

   if 6.8000000000000003e67 < l

   1. Initial program 36.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 42.7%

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

   10. Applied rewrites 64.0%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot n}{Om} \cdot \left(\ell \cdot \ell\right), -2, \left(t \cdot n\right) \cdot 2\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 145000000:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 145000000.0)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (* (* (* -2.0 U) l) (* (/ l Om) n)) (fma (/ n Om) (- U U*) 2.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 145000000.0) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * ((l / Om) * n)) * fma((n / Om), (U - U_42_), 2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 145000000.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(Float64(l / Om) * n)) * fma(Float64(n / Om), Float64(U - U_42_), 2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 145000000.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.45e8

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 45.4

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

   5. Applied rewrites 45.4%

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

   if 1.45e8 < l

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 45.3%

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

   10. Applied rewrites 64.2%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 145000000:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.2e+52)
   (sqrt (* (* (* (fma -2.0 (/ (* l l) Om) t) n) U) 2.0))
   (sqrt (* (* (* (* -2.0 U) l) (fma (/ n Om) (- U U*) 2.0)) (* (/ l Om) n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.2e+52) {
		tmp = sqrt((((fma(-2.0, ((l * l) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * fma((n / Om), (U - U_42_), 2.0)) * ((l / Om) * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.2e+52)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * fma(Float64(n / Om), Float64(U - U_42_), 2.0)) * Float64(Float64(l / Om) * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.2e+52], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.2e52

   1. Initial program 52.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.6

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

   5. Applied rewrites 48.6%

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

   if 7.2e52 < l

   1. Initial program 37.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 45.1%

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

   10. Applied rewrites 65.5%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;n \leq 60000000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.2e+146)
   (sqrt (* (* (* t U) n) 2.0))
   (if (<= n 60000000000000.0)
     (sqrt (fma (/ (* (* (* l n) l) U) Om) -4.0 (* (* (* t n) U) 2.0)))
     (* (sqrt (* (* (fma (* (/ l Om) l) -2.0 t) U) 2.0)) (sqrt n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.2e+146) {
		tmp = sqrt((((t * U) * n) * 2.0));
	} else if (n <= 60000000000000.0) {
		tmp = sqrt(fma(((((l * n) * l) * U) / Om), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) * U) * 2.0)) * sqrt(n);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.2e+146)
		tmp = sqrt(Float64(Float64(Float64(t * U) * n) * 2.0));
	elseif (n <= 60000000000000.0)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l * n) * l) * U) / Om), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * U) * 2.0)) * sqrt(n));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.2e+146], N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 60000000000000.0], N[Sqrt[N[(N[(N[(N[(N[(l * n), $MachinePrecision] * l), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2000000000000001e146

   1. Initial program 47.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 40.4

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

   5. Applied rewrites 40.4%

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

   7. Applied rewrites 46.5%

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

   if -1.2000000000000001e146 < n < 6e13

   1. Initial program 48.4%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   7. Applied rewrites 56.5%

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

   if 6e13 < n

   1. Initial program 53.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 53.7

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

   7. Applied rewrites 53.7%

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

   9. Applied rewrites 63.9%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;n \leq 60000000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot n} \cdot \sqrt{t}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.25e+171)
   (sqrt (fma (/ (* (* (* l n) l) U) Om) -4.0 (* (* (* t n) U) 2.0)))
   (* (* (sqrt (* U n)) (sqrt t)) (sqrt 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.25e+171) {
		tmp = sqrt(fma(((((l * n) * l) * U) / Om), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = (sqrt((U * n)) * sqrt(t)) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.25e+171)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l * n) * l) * U) / Om), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(U * n)) * sqrt(t)) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.25e+171], N[Sqrt[N[(N[(N[(N[(N[(l * n), $MachinePrecision] * l), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2500000000000001e171

   1. Initial program 50.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 51.9%

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

   if 1.2500000000000001e171 < t

   1. Initial program 40.7%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 61.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot n} \cdot \sqrt{t}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{n}{Om} \cdot \ell\right) \cdot \left(\ell \cdot U\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot n} \cdot \sqrt{t}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.25e+171)
   (sqrt (fma (* (* (/ n Om) l) (* l U)) -4.0 (* (* (* t n) U) 2.0)))
   (* (* (sqrt (* U n)) (sqrt t)) (sqrt 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.25e+171) {
		tmp = sqrt(fma((((n / Om) * l) * (l * U)), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = (sqrt((U * n)) * sqrt(t)) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.25e+171)
		tmp = sqrt(fma(Float64(Float64(Float64(n / Om) * l) * Float64(l * U)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(U * n)) * sqrt(t)) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.25e+171], N[Sqrt[N[(N[(N[(N[(n / Om), $MachinePrecision] * l), $MachinePrecision] * N[(l * U), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2500000000000001e171

   1. Initial program 50.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 51.6%

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

   if 1.2500000000000001e171 < t

   1. Initial program 40.7%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 61.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{n}{Om} \cdot \ell\right) \cdot \left(\ell \cdot U\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot n} \cdot \sqrt{t}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot n} \cdot \sqrt{t}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.25e+171)
   (sqrt (* (* (* (fma -2.0 (/ (* l l) Om) t) n) U) 2.0))
   (* (* (sqrt (* U n)) (sqrt t)) (sqrt 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.25e+171) {
		tmp = sqrt((((fma(-2.0, ((l * l) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = (sqrt((U * n)) * sqrt(t)) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.25e+171)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * n) * U) * 2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(U * n)) * sqrt(t)) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.25e+171], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2500000000000001e171

   1. Initial program 50.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   if 1.2500000000000001e171 < t

   1. Initial program 40.7%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 61.9%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot n} \cdot \sqrt{t}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.6e+57)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (/ (* (* (* l l) n) U) Om) -4.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.6e+57) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((l * l) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.6d+57) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((((((l * l) * n) * u) / om) * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.6e+57) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((((l * l) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.6e+57:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((((((l * l) * n) * U) / Om) * -4.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.6e+57)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * U) / Om) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.6e+57)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((((((l * l) * n) * U) / Om) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.6e+57], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.6000000000000002e57

   1. Initial program 52.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   if 6.6000000000000002e57 < l

   1. Initial program 35.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 43.9%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 34.9%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 200000000000:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 200000000000.0)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* (* t U) 2.0)) (sqrt n))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 200000000000.0) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(((t * U) * 2.0)) * sqrt(n);
	}
	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 <= 200000000000.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((t * u) * 2.0d0)) * sqrt(n)
    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 <= 200000000000.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((t * U) * 2.0)) * Math.sqrt(n);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 200000000000.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt(((t * U) * 2.0)) * math.sqrt(n)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 200000000000.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(t * U) * 2.0)) * sqrt(n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 200000000000.0)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt(((t * U) * 2.0)) * sqrt(n);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 200000000000.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2e11

   1. Initial program 48.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 41.7

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

   5. Applied rewrites 41.7%

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

   if 2e11 < n

   1. Initial program 53.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   4. Applied rewrites 43.4%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 37.8

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

   7. Applied rewrites 37.8%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 200000000000:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -4.2 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -4.2e+91)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* (* (* t n) U) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -4.2e+91) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((((t * n) * U) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= (-4.2d+91)) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((((t * n) * u) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -4.2e+91) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -4.2e+91:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -4.2e+91)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -4.2e+91)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((((t * n) * U) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -4.2e+91], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -4.20000000000000015e91

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   7. Applied rewrites 45.8%

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

   if -4.20000000000000015e91 < U*

   1. Initial program 47.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 38.3

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

   5. Applied rewrites 38.3%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -4.2 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 38.2

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

5. Applied rewrites 38.2%

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

7. Applied rewrites 36.8%

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

8. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2024244 
(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*))))))