Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 65.2%

Time: 7.9s

Alternatives: 15

Speedup: 2.3×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    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: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U - U*, \ell \cdot 2\right)\\ \mathbf{if}\;U \leq 1.75 \cdot 10^{-294}:\\ \;\;\;\;{\left(t\_1 \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot t\_1} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ l Om) (fma (* (/ l Om) n) (- U U*) (* l 2.0))))))
   (if (<= U 1.75e-294)
     (pow (* t_1 (* (* U 2.0) n)) 0.5)
     (* (sqrt (* (* n 2.0) t_1)) (sqrt U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * fma(((l / Om) * n), (U - U_42_), (l * 2.0)));
	double tmp;
	if (U <= 1.75e-294) {
		tmp = pow((t_1 * ((U * 2.0) * n)), 0.5);
	} else {
		tmp = sqrt(((n * 2.0) * t_1)) * sqrt(U);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l / Om) * fma(Float64(Float64(l / Om) * n), Float64(U - U_42_), Float64(l * 2.0))))
	tmp = 0.0
	if (U <= 1.75e-294)
		tmp = Float64(t_1 * Float64(Float64(U * 2.0) * n)) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(Float64(n * 2.0) * t_1)) * sqrt(U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, 1.75e-294], N[Power[N[(t$95$1 * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 1.75000000000000016e-294

   1. Initial program 48.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 54.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 51.0

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

   3. Applied rewrites 51.0%

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

   5. Applied rewrites 59.5%

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

   if 1.75000000000000016e-294 < U

   1. Initial program 52.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 57.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 54.5

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

   3. Applied rewrites 54.5%

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

   5. Applied rewrites 71.1%

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

Alternative 2: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{Om} \cdot n\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 10^{+58}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(t\_1, U - U*, \ell \cdot 2\right)\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(t\_1, U - U*, \ell + \ell\right)\right) \cdot U\right) \cdot \left(n + n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(2, \ell, \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l Om) n))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 1e+58)
     (sqrt (* (* (- t (* (/ l Om) (fma t_1 (- U U*) (* l 2.0)))) (* n 2.0)) U))
     (if (<= t_2 2e+154)
       (sqrt (* (* (+ n n) U) (- t (* (* n (/ (* l l) (* Om Om))) (- U U*)))))
       (if (<= t_2 INFINITY)
         (sqrt (* (* (- t (* (/ l Om) (fma t_1 (- U U*) (+ l l)))) U) (+ n n)))
         (sqrt
          (*
           -2.0
           (/
            (* U (* l (* n (fma 2.0 l (/ (* l (* n (- U U*))) Om)))))
            Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * n;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e+58) {
		tmp = sqrt((((t - ((l / Om) * fma(t_1, (U - U_42_), (l * 2.0)))) * (n * 2.0)) * U));
	} else if (t_2 <= 2e+154) {
		tmp = sqrt((((n + n) * U) * (t - ((n * ((l * l) / (Om * Om))) * (U - U_42_)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((((t - ((l / Om) * fma(t_1, (U - U_42_), (l + l)))) * U) * (n + n)));
	} else {
		tmp = sqrt((-2.0 * ((U * (l * (n * fma(2.0, l, ((l * (n * (U - U_42_))) / Om))))) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l / Om) * n)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 1e+58)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(l / Om) * fma(t_1, Float64(U - U_42_), Float64(l * 2.0)))) * Float64(n * 2.0)) * U));
	elseif (t_2 <= 2e+154)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l * l) / Float64(Om * Om))) * Float64(U - U_42_)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(l / Om) * fma(t_1, Float64(U - U_42_), Float64(l + l)))) * U) * Float64(n + n)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(l * Float64(n * fma(2.0, l, Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om))))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+58], N[Sqrt[N[(N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(t$95$1 * N[(U - U$42$), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+154], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(t$95$1 * N[(U - U$42$), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * N[(n * N[(2.0 * l + N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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*))))) < 9.99999999999999944e57

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 74.1%

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

   5. Applied rewrites 75.0%

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

   if 9.99999999999999944e57 < (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*))))) < 2.00000000000000007e154

   1. Initial program 99.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 87.0

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

   3. Applied rewrites 87.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 81.8%

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

   if 2.00000000000000007e154 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 32.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 42.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 39.4

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

   3. Applied rewrites 39.4%

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

   5. Applied rewrites 43.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 43.6

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

   7. Applied rewrites 43.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 43.6

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

   9. Applied rewrites 43.6%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 14.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 14.6

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

   3. Applied rewrites 14.6%

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 54.5%

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

Alternative 3: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U - U*, \ell + \ell\right)\right) \cdot U\right) \cdot \left(n + n\right)}\\ \mathbf{if}\;t\_1 \leq 7 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(2, \ell, \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2
         (sqrt
          (*
           (* (- t (* (/ l Om) (fma (* (/ l Om) n) (- U U*) (+ l l)))) U)
           (+ n n)))))
   (if (<= t_1 7e-16)
     t_2
     (if (<= t_1 4e+148)
       (sqrt
        (* (- t (* (/ l Om) (* -1.0 (/ (* U* (* l n)) Om)))) (* (* U 2.0) n)))
       (if (<= t_1 INFINITY)
         t_2
         (sqrt
          (*
           -2.0
           (/
            (* U (* l (* n (fma 2.0 l (/ (* l (* n (- U U*))) Om)))))
            Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt((((t - ((l / Om) * fma(((l / Om) * n), (U - U_42_), (l + l)))) * U) * (n + n)));
	double tmp;
	if (t_1 <= 7e-16) {
		tmp = t_2;
	} else if (t_1 <= 4e+148) {
		tmp = sqrt(((t - ((l / Om) * (-1.0 * ((U_42_ * (l * n)) / Om)))) * ((U * 2.0) * n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = sqrt((-2.0 * ((U * (l * (n * fma(2.0, l, ((l * (n * (U - U_42_))) / Om))))) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	t_2 = sqrt(Float64(Float64(Float64(t - Float64(Float64(l / Om) * fma(Float64(Float64(l / Om) * n), Float64(U - U_42_), Float64(l + l)))) * U) * Float64(n + n)))
	tmp = 0.0
	if (t_1 <= 7e-16)
		tmp = t_2;
	elseif (t_1 <= 4e+148)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(l / Om) * Float64(-1.0 * Float64(Float64(U_42_ * Float64(l * n)) / Om)))) * Float64(Float64(U * 2.0) * n)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(l * Float64(n * fma(2.0, l, Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om))))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 7e-16], t$95$2, If[LessEqual[t$95$1, 4e+148], N[Sqrt[N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(-1.0 * N[(N[(U$42$ * N[(l * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * N[(n * N[(2.0 * l + N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 7.00000000000000035e-16 or 4.0000000000000002e148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 46.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 51.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 48.6

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

   3. Applied rewrites 48.6%

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

   5. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 55.6

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

   7. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 55.6

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

   9. Applied rewrites 55.6%

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

   if 7.00000000000000035e-16 < (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*))))) < 4.0000000000000002e148

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 98.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 91.1

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

   3. Applied rewrites 91.1%

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 14.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 14.6

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

   3. Applied rewrites 14.6%

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 54.5%

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

Alternative 4: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U - U*, \ell \cdot 2\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(2, \ell, \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ l Om) (fma (* (/ l Om) n) (- U U*) (* l 2.0)))))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 4e-109)
     (sqrt (* (* t_1 (* n 2.0)) U))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (* (* U 2.0) n)))
       (sqrt
        (*
         -2.0
         (/ (* U (* l (* n (fma 2.0 l (/ (* l (* n (- U U*))) Om))))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * fma(((l / Om) * n), (U - U_42_), (l * 2.0)));
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 4e-109) {
		tmp = sqrt(((t_1 * (n * 2.0)) * U));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((U * 2.0) * n)));
	} else {
		tmp = sqrt((-2.0 * ((U * (l * (n * fma(2.0, l, ((l * (n * (U - U_42_))) / Om))))) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l / Om) * fma(Float64(Float64(l / Om) * n), Float64(U - U_42_), Float64(l * 2.0))))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 4e-109)
		tmp = sqrt(Float64(Float64(t_1 * Float64(n * 2.0)) * U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(U * 2.0) * n)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(l * Float64(n * fma(2.0, l, Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om))))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4e-109], N[Sqrt[N[(N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * N[(n * N[(2.0 * l + N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4e-109

   1. Initial program 31.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 47.8%

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

   5. Applied rewrites 48.4%

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

   if 4e-109 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 68.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 72.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.0

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

   3. Applied rewrites 68.0%

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

   5. Applied rewrites 73.7%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 14.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 14.6

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

   3. Applied rewrites 14.6%

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 54.5%

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

Alternative 5: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U - U*\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell \cdot \mathsf{fma}\left(2, \ell, \frac{U \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot \left(2 + \frac{t\_1}{Om}\right)\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(2, \ell, \frac{\ell \cdot t\_1}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- U U*)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 4e-109)
     (sqrt
      (* 2.0 (* U (* n (- t (/ (* l (fma 2.0 l (/ (* U (* l n)) Om))) Om))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (- t (* (/ l Om) (* l (+ 2.0 (/ t_1 Om))))) (* (* U 2.0) n)))
       (sqrt
        (* -2.0 (/ (* U (* l (* n (fma 2.0 l (/ (* l t_1) Om))))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U - U_42_);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 4e-109) {
		tmp = sqrt((2.0 * (U * (n * (t - ((l * fma(2.0, l, ((U * (l * n)) / Om))) / Om))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((t - ((l / Om) * (l * (2.0 + (t_1 / Om))))) * ((U * 2.0) * n)));
	} else {
		tmp = sqrt((-2.0 * ((U * (l * (n * fma(2.0, l, ((l * t_1) / Om))))) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U - U_42_))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 4e-109)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(Float64(l * fma(2.0, l, Float64(Float64(U * Float64(l * n)) / Om))) / Om))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(l / Om) * Float64(l * Float64(2.0 + Float64(t_1 / Om))))) * Float64(Float64(U * 2.0) * n)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(l * Float64(n * fma(2.0, l, Float64(Float64(l * t_1) / Om))))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4e-109], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(N[(l * N[(2.0 * l + N[(N[(U * N[(l * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(l * N[(2.0 + N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * N[(n * N[(2.0 * l + N[(N[(l * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4e-109

   1. Initial program 31.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 31.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 31.3

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

   3. Applied rewrites 31.3%

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in U* around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 4e-109 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 68.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 72.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.0

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

   3. Applied rewrites 68.0%

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 66.5

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

   8. Applied rewrites 66.5%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 14.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 14.6

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

   3. Applied rewrites 14.6%

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 54.5%

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

Alternative 6: 53.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ l Om) (* 2.0 l))))
        (t_2
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 2e-310)
     (sqrt (* (* t_1 U) (* n 2.0)))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (* (* U 2.0) n)))
       (* (/ (* l (* n (sqrt 2.0))) Om) (sqrt (* U U*)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * (2.0 * l));
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 2e-310) {
		tmp = sqrt(((t_1 * U) * (n * 2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((U * 2.0) * n)));
	} else {
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * (2.0 * l));
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 2e-310) {
		tmp = Math.sqrt(((t_1 * U) * (n * 2.0)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * ((U * 2.0) * n)));
	} else {
		tmp = ((l * (n * Math.sqrt(2.0))) / Om) * Math.sqrt((U * U_42_));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - ((l / Om) * (2.0 * l))
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 2e-310:
		tmp = math.sqrt(((t_1 * U) * (n * 2.0)))
	elif t_2 <= math.inf:
		tmp = math.sqrt((t_1 * ((U * 2.0) * n)))
	else:
		tmp = ((l * (n * math.sqrt(2.0))) / Om) * math.sqrt((U * U_42_))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 2e-310)
		tmp = sqrt(Float64(Float64(t_1 * U) * Float64(n * 2.0)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(U * 2.0) * n)));
	else
		tmp = Float64(Float64(Float64(l * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - ((l / Om) * (2.0 * l));
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 2e-310)
		tmp = sqrt(((t_1 * U) * (n * 2.0)));
	elseif (t_2 <= Inf)
		tmp = sqrt((t_1 * ((U * 2.0) * n)));
	else
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-310], N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $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*)))) < 1.999999999999994e-310

   1. Initial program 12.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 14.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 15.1

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

   3. Applied rewrites 15.1%

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

   5. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 41.8

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

   7. Applied rewrites 41.8%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 37.4

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

   10. Applied rewrites 37.4%

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

   if 1.999999999999994e-310 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 69.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 74.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 69.4

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

   3. Applied rewrites 69.4%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 63.9

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

   8. Applied rewrites 63.9%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 13.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 13.2

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

   3. Applied rewrites 13.2%

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 23.3

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

   10. Applied rewrites 23.3%

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

Alternative 7: 39.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (* 2.0 (* U (* n t))))))
   (if (<= t_1 5e-142)
     t_2
     (if (<= t_1 2e+145) (sqrt (* (* (+ n n) U) t)) t_2))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt((2.0 * (U * (n * t))));
	double tmp;
	if (t_1 <= 5e-142) {
		tmp = t_2;
	} else if (t_1 <= 2e+145) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    t_2 = sqrt((2.0d0 * (u * (n * t))))
    if (t_1 <= 5d-142) then
        tmp = t_2
    else if (t_1 <= 2d+145) then
        tmp = sqrt((((n + n) * u) * t))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = Math.sqrt((2.0 * (U * (n * t))));
	double tmp;
	if (t_1 <= 5e-142) {
		tmp = t_2;
	} else if (t_1 <= 2e+145) {
		tmp = Math.sqrt((((n + n) * U) * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	t_2 = math.sqrt((2.0 * (U * (n * t))))
	tmp = 0
	if t_1 <= 5e-142:
		tmp = t_2
	elif t_1 <= 2e+145:
		tmp = math.sqrt((((n + n) * U) * t))
	else:
		tmp = t_2
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	t_2 = sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
	tmp = 0.0
	if (t_1 <= 5e-142)
		tmp = t_2;
	elseif (t_1 <= 2e+145)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	t_2 = sqrt((2.0 * (U * (n * t))));
	tmp = 0.0;
	if (t_1 <= 5e-142)
		tmp = t_2;
	elseif (t_1 <= 2e+145)
		tmp = sqrt((((n + n) * U) * t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e-142], t$95$2, If[LessEqual[t$95$1, 2e+145], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

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*))))) < 5.0000000000000002e-142 or 2e145 < (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 21.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 29.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 28.5

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

   3. Applied rewrites 28.5%

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 17.6

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

   8. Applied rewrites 17.6%

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

   if 5.0000000000000002e-142 < (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*))))) < 2e145

   1. Initial program 98.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 86.6

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

   3. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.5%

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

Alternative 8: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U - U*, \ell \cdot 2\right)\\ \mathbf{if}\;U \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot t\_1} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ l Om) (fma (* (/ l Om) n) (- U U*) (* l 2.0))))))
   (if (<= U -5e-311)
     (sqrt (* t_1 (* (* U 2.0) n)))
     (* (sqrt (* (* n 2.0) t_1)) (sqrt U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * fma(((l / Om) * n), (U - U_42_), (l * 2.0)));
	double tmp;
	if (U <= -5e-311) {
		tmp = sqrt((t_1 * ((U * 2.0) * n)));
	} else {
		tmp = sqrt(((n * 2.0) * t_1)) * sqrt(U);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l / Om) * fma(Float64(Float64(l / Om) * n), Float64(U - U_42_), Float64(l * 2.0))))
	tmp = 0.0
	if (U <= -5e-311)
		tmp = sqrt(Float64(t_1 * Float64(Float64(U * 2.0) * n)));
	else
		tmp = Float64(sqrt(Float64(Float64(n * 2.0) * t_1)) * sqrt(U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -5e-311], N[Sqrt[N[(t$95$1 * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -5.00000000000023e-311

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 54.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 51.7

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

   3. Applied rewrites 51.7%

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

   5. Applied rewrites 60.1%

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

   if -5.00000000000023e-311 < U

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 56.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 53.8

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

   3. Applied rewrites 53.8%

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

   5. Applied rewrites 70.7%

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

Alternative 9: 54.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{if}\;Om \leq -2.2 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (- t (* (/ l Om) (* 2.0 l))) (* (* U 2.0) n)))))
   (if (<= Om -2.2e-63)
     t_1
     (if (<= Om 1.6e+64)
       (sqrt
        (* (* (- t (* (/ l Om) (/ (* l (* n (- U U*))) Om))) U) (* n 2.0)))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	double tmp;
	if (Om <= -2.2e-63) {
		tmp = t_1;
	} else if (Om <= 1.6e+64) {
		tmp = sqrt((((t - ((l / Om) * ((l * (n * (U - U_42_))) / Om))) * U) * (n * 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((t - ((l / om) * (2.0d0 * l))) * ((u * 2.0d0) * n)))
    if (om <= (-2.2d-63)) then
        tmp = t_1
    else if (om <= 1.6d+64) then
        tmp = sqrt((((t - ((l / om) * ((l * (n * (u - u_42))) / om))) * u) * (n * 2.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	double tmp;
	if (Om <= -2.2e-63) {
		tmp = t_1;
	} else if (Om <= 1.6e+64) {
		tmp = Math.sqrt((((t - ((l / Om) * ((l * (n * (U - U_42_))) / Om))) * U) * (n * 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)))
	tmp = 0
	if Om <= -2.2e-63:
		tmp = t_1
	elif Om <= 1.6e+64:
		tmp = math.sqrt((((t - ((l / Om) * ((l * (n * (U - U_42_))) / Om))) * U) * (n * 2.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))) * Float64(Float64(U * 2.0) * n)))
	tmp = 0.0
	if (Om <= -2.2e-63)
		tmp = t_1;
	elseif (Om <= 1.6e+64)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(l / Om) * Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om))) * U) * Float64(n * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	tmp = 0.0;
	if (Om <= -2.2e-63)
		tmp = t_1;
	elseif (Om <= 1.6e+64)
		tmp = sqrt((((t - ((l / Om) * ((l * (n * (U - U_42_))) / Om))) * U) * (n * 2.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -2.2e-63], t$95$1, If[LessEqual[Om, 1.6e+64], N[Sqrt[N[(N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.2e-63 or 1.60000000000000009e64 < Om

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 60.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 55.8

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

   3. Applied rewrites 55.8%

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

   if -2.2e-63 < Om < 1.60000000000000009e64

   1. Initial program 46.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 48.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 48.5

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

   3. Applied rewrites 48.5%

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

   5. Applied rewrites 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 59.7

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

   7. Applied rewrites 59.7%

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

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 53.9

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

   10. Applied rewrites 53.9%

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

Alternative 10: 61.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, U - U*, \ell + \ell\right)\right) \cdot U\right) \cdot \left(n + n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{U \cdot \left(n + n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2.5e+179)
   (sqrt
    (* (* (- t (* (/ l Om) (fma (* (/ l Om) n) (- U U*) (+ l l)))) U) (+ n n)))
   (* (sqrt t) (sqrt (* U (+ n n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.5e+179) {
		tmp = sqrt((((t - ((l / Om) * fma(((l / Om) * n), (U - U_42_), (l + l)))) * U) * (n + n)));
	} else {
		tmp = sqrt(t) * sqrt((U * (n + n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2.5e+179)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(l / Om) * fma(Float64(Float64(l / Om) * n), Float64(U - U_42_), Float64(l + l)))) * U) * Float64(n + n)));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(U * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2.5e+179], N[Sqrt[N[(N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e179

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 56.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 53.2

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

   3. Applied rewrites 53.2%

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

   5. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 61.1

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

   7. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 61.1

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

   9. Applied rewrites 61.1%

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

   if 2.5e179 < t

   1. Initial program 49.6%

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

      1. 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. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 59.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 54.0

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

   5. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 54.0

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 61.3%

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

Alternative 11: 50.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{if}\;Om \leq -1.52 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 3.3 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (- t (* (/ l Om) (* 2.0 l))) (* (* U 2.0) n)))))
   (if (<= Om -1.52e+54)
     t_1
     (if (<= Om 3.3e+65)
       (sqrt (* (* (+ n n) U) (- t (* (* n (/ (* l l) (* Om Om))) (- U U*)))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	double tmp;
	if (Om <= -1.52e+54) {
		tmp = t_1;
	} else if (Om <= 3.3e+65) {
		tmp = sqrt((((n + n) * U) * (t - ((n * ((l * l) / (Om * Om))) * (U - U_42_)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((t - ((l / om) * (2.0d0 * l))) * ((u * 2.0d0) * n)))
    if (om <= (-1.52d+54)) then
        tmp = t_1
    else if (om <= 3.3d+65) then
        tmp = sqrt((((n + n) * u) * (t - ((n * ((l * l) / (om * om))) * (u - u_42)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	double tmp;
	if (Om <= -1.52e+54) {
		tmp = t_1;
	} else if (Om <= 3.3e+65) {
		tmp = Math.sqrt((((n + n) * U) * (t - ((n * ((l * l) / (Om * Om))) * (U - U_42_)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)))
	tmp = 0
	if Om <= -1.52e+54:
		tmp = t_1
	elif Om <= 3.3e+65:
		tmp = math.sqrt((((n + n) * U) * (t - ((n * ((l * l) / (Om * Om))) * (U - U_42_)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))) * Float64(Float64(U * 2.0) * n)))
	tmp = 0.0
	if (Om <= -1.52e+54)
		tmp = t_1;
	elseif (Om <= 3.3e+65)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l * l) / Float64(Om * Om))) * Float64(U - U_42_)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	tmp = 0.0;
	if (Om <= -1.52e+54)
		tmp = t_1;
	elseif (Om <= 3.3e+65)
		tmp = sqrt((((n + n) * U) * (t - ((n * ((l * l) / (Om * Om))) * (U - U_42_)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -1.52e+54], t$95$1, If[LessEqual[Om, 3.3e+65], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.51999999999999999e54 or 3.30000000000000023e65 < Om

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 61.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 55.8

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

   3. Applied rewrites 55.8%

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 57.9

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

   8. Applied rewrites 57.9%

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

   if -1.51999999999999999e54 < Om < 3.30000000000000023e65

   1. Initial program 47.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 38.8

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

   3. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 43.8%

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

Alternative 12: 47.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{U \cdot \left(n + n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.08e+42)
   (sqrt (* (- t (* (/ l Om) (* 2.0 l))) (* (* U 2.0) n)))
   (* (sqrt t) (sqrt (* U (+ n n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.08e+42) {
		tmp = sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	} else {
		tmp = sqrt(t) * sqrt((U * (n + n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.08d+42) then
        tmp = sqrt(((t - ((l / om) * (2.0d0 * l))) * ((u * 2.0d0) * n)))
    else
        tmp = sqrt(t) * sqrt((u * (n + 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 (t <= 1.08e+42) {
		tmp = Math.sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((U * (n + n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.08e+42:
		tmp = math.sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)))
	else:
		tmp = math.sqrt(t) * math.sqrt((U * (n + n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.08e+42)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))) * Float64(Float64(U * 2.0) * n)));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(U * Float64(n + n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.08e+42)
		tmp = sqrt(((t - ((l / Om) * (2.0 * l))) * ((U * 2.0) * n)));
	else
		tmp = sqrt(t) * sqrt((U * (n + n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.08e+42], N[Sqrt[N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.08e42

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 55.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 52.8

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

   3. Applied rewrites 52.8%

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 46.2

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

   8. Applied rewrites 46.2%

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

   if 1.08e42 < t

   1. Initial program 51.8%

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

      1. 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. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 56.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 50.6

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

   5. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 50.6

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

   7. Applied rewrites 50.6%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 53.9%

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

Alternative 13: 48.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{U \cdot \left(n + n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 7e+159)
   (sqrt (* (* (- t (* (/ l Om) (* 2.0 l))) U) (* n 2.0)))
   (* (sqrt t) (sqrt (* U (+ n n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 7e+159) {
		tmp = sqrt((((t - ((l / Om) * (2.0 * l))) * U) * (n * 2.0)));
	} else {
		tmp = sqrt(t) * sqrt((U * (n + n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 7d+159) then
        tmp = sqrt((((t - ((l / om) * (2.0d0 * l))) * u) * (n * 2.0d0)))
    else
        tmp = sqrt(t) * sqrt((u * (n + 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 (t <= 7e+159) {
		tmp = Math.sqrt((((t - ((l / Om) * (2.0 * l))) * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((U * (n + n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 7e+159:
		tmp = math.sqrt((((t - ((l / Om) * (2.0 * l))) * U) * (n * 2.0)))
	else:
		tmp = math.sqrt(t) * math.sqrt((U * (n + n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 7e+159)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))) * U) * Float64(n * 2.0)));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(U * Float64(n + n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 7e+159)
		tmp = sqrt((((t - ((l / Om) * (2.0 * l))) * U) * (n * 2.0)));
	else
		tmp = sqrt(t) * sqrt((U * (n + n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 7e+159], N[Sqrt[N[(N[(N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.9999999999999999e159

   1. Initial program 50.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 56.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 53.2

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

   3. Applied rewrites 53.2%

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

   5. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 61.0

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 47.2

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

   10. Applied rewrites 47.2%

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

   if 6.9999999999999999e159 < t

   1. Initial program 50.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 59.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 53.4

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 60.0%

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

Alternative 14: 39.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{U \cdot \left(n + n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -8e-298)
   (sqrt (* (* (+ n n) U) t))
   (* (sqrt t) (sqrt (* U (+ n n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -8e-298) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = sqrt(t) * sqrt((U * (n + n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-8d-298)) then
        tmp = sqrt((((n + n) * u) * t))
    else
        tmp = sqrt(t) * sqrt((u * (n + 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 (t <= -8e-298) {
		tmp = Math.sqrt((((n + n) * U) * t));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((U * (n + n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -8e-298:
		tmp = math.sqrt((((n + n) * U) * t))
	else:
		tmp = math.sqrt(t) * math.sqrt((U * (n + n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -8e-298)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(U * Float64(n + n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -8e-298)
		tmp = sqrt((((n + n) * U) * t));
	else
		tmp = sqrt(t) * sqrt((U * (n + n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -8e-298], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.9999999999999993e-298

   1. Initial program 50.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 44.8

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

   3. Applied rewrites 44.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 44.8

         \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 44.8%

      \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -7.9999999999999993e-298 < t

   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. 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. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}^{\frac{1}{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}^{\frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)}}^{\frac{1}{2}} \]

      5. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{\frac{1}{2}} \cdot {\left(\left(2 \cdot n\right) \cdot U\right)}^{\frac{1}{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{\frac{1}{2}} \cdot {\left(\left(2 \cdot n\right) \cdot U\right)}^{\frac{1}{2}}} \]

   3. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{2} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)}} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\color{blue}{\frac{\ell}{Om}} \cdot \frac{\ell}{Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\frac{\ell}{Om} \cdot \color{blue}{\frac{\ell}{Om}}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      5. times-frac N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \color{blue}{\frac{\ell \cdot \ell}{Om \cdot Om}} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      6. sqr-neg-rev N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \frac{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)}}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \frac{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)}{\color{blue}{Om \cdot Om}} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \frac{\mathsf{neg}\left(\ell\right)}{Om \cdot Om}\right)} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \frac{\mathsf{neg}\left(\ell\right)}{Om \cdot Om}\right)} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\color{blue}{\left(-\ell\right)} \cdot \frac{\mathsf{neg}\left(\ell\right)}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\ell\right)}{Om \cdot Om}}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

      12. lower-neg.f64 46.1

          \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \frac{\color{blue}{-\ell}}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]

   5. Applied rewrites 46.1%

      \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \color{blue}{\left(\left(-\ell\right) \cdot \frac{-\ell}{Om \cdot Om}\right)} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n \cdot 2\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \frac{-\ell}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \color{blue}{\left(n \cdot 2\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \frac{-\ell}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \color{blue}{\left(2 \cdot n\right)}} \]

      3. count-2-rev N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \frac{-\ell}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \color{blue}{\left(n + n\right)}} \]

      4. lower-+.f64 46.1

         \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \frac{-\ell}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \color{blue}{\left(n + n\right)}} \]

   7. Applied rewrites 46.1%

      \[\leadsto \sqrt{\mathsf{fma}\left(U* - U, \left(\left(-\ell\right) \cdot \frac{-\ell}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \color{blue}{\left(n + n\right)}} \]

   8. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{t}} \cdot \sqrt{U \cdot \left(n + n\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 43.0%

       \[\leadsto \sqrt{\color{blue}{t}} \cdot \sqrt{U \cdot \left(n + n\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* (+ n n) U) t)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((n + n) * U) * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    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((((n + 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((((n + n) * U) * t));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((n + n) * U) * t))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(n + n) * U) * t))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((n + n) * U) * t));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\color{blue}{\frac{\ell}{Om}} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\frac{\ell}{Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. frac-times N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{\ell \cdot \ell}{Om \cdot Om}}\right) \cdot \left(U - U*\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{\ell \cdot \ell}{Om \cdot Om}}\right) \cdot \left(U - U*\right)\right)} \]

   8. lower-*.f64 45.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right) \cdot \left(U - U*\right)\right)} \]

3. Applied rewrites 45.1%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{\ell \cdot \ell}{Om \cdot Om}}\right) \cdot \left(U - U*\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

   2. count-2-rev N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

   3. lower-+.f64 45.1

      \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

5. Applied rewrites 45.1%

   \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 36.5%

   \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025108 
(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*))))))