Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 64.8%

Time: 11.6s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

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: 50.2% accurate, 1.0× speedup

Math

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

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: 64.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := -\left|\ell\right|\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \left(n \cdot {\left(\frac{\left|\ell\right|}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_4 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - t\_3\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\left(U - U*\right) \cdot n, \frac{\left|\ell\right|}{Om \cdot Om}, \frac{\left|\ell\right| + \left|\ell\right|}{Om}\right) \cdot \left|\ell\right|\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - t\_1 \cdot \left(t\_1 \cdot \frac{2}{Om}\right)\right) - t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- (fabs l)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* (* n (pow (/ (fabs l) Om) 2.0)) (- U U*)))
        (t_4
         (sqrt (* t_2 (- (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om))) t_3)))))
   (if (<= t_4 0.0)
     (*
      (sqrt
       (*
        (-
         t
         (*
          (fma
           (* (- U U*) n)
           (/ (fabs l) (* Om Om))
           (/ (+ (fabs l) (fabs l)) Om))
          (fabs l)))
        (+ n n)))
      (sqrt U))
     (if (<= t_4 INFINITY)
       (sqrt (* t_2 (- (- t (* t_1 (* t_1 (/ 2.0 Om)))) t_3)))
       (*
        (fabs l)
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = -fabs(l);
	double t_2 = (2.0 * n) * U;
	double t_3 = (n * pow((fabs(l) / Om), 2.0)) * (U - U_42_);
	double t_4 = sqrt((t_2 * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((t - (fma(((U - U_42_) * n), (fabs(l) / (Om * Om)), ((fabs(l) + fabs(l)) / Om)) * fabs(l))) * (n + n))) * sqrt(U);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (t_1 * (t_1 * (2.0 / Om)))) - t_3)));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-abs(l))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(n * (Float64(abs(l) / Om) ^ 2.0)) * Float64(U - U_42_))
	t_4 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t - Float64(fma(Float64(Float64(U - U_42_) * n), Float64(abs(l) / Float64(Om * Om)), Float64(Float64(abs(l) + abs(l)) / Om)) * abs(l))) * Float64(n + n))) * sqrt(U));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(t_1 * Float64(t_1 * Float64(2.0 / Om)))) - t_3)));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = (-N[Abs[l], $MachinePrecision])}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * N[Power[N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(t$95$1 * N[(t$95$1 * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 27.4%

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

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #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 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. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.5%

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

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 64.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right|}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {t\_1}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\left(U - U*\right) \cdot n, \frac{\left|\ell\right|}{Om \cdot Om}, \frac{\left|\ell\right| + \left|\ell\right|}{Om}\right) \cdot \left|\ell\right|\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{-2 \cdot \left|\ell\right|}{Om}, \left|\ell\right|, t\right) - \left(n \cdot \frac{t\_1 \cdot \left|\ell\right|}{Om}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (fabs l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
            (* (* n (pow t_1 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (*
      (sqrt
       (*
        (-
         t
         (*
          (fma
           (* (- U U*) n)
           (/ (fabs l) (* Om Om))
           (/ (+ (fabs l) (fabs l)) Om))
          (fabs l)))
        (+ n n)))
      (sqrt U))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (fma (/ (* -2.0 (fabs l)) Om) (fabs l) t)
          (* (* n (/ (* t_1 (fabs l)) Om)) (- U U*)))))
       (*
        (fabs l)
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(l) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow(t_1, 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((t - (fma(((U - U_42_) * n), (fabs(l) / (Om * Om)), ((fabs(l) + fabs(l)) / Om)) * fabs(l))) * (n + n))) * sqrt(U);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((((n + n) * U) * (fma(((-2.0 * fabs(l)) / Om), fabs(l), t) - ((n * ((t_1 * fabs(l)) / Om)) * (U - U_42_)))));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Abs[l], $MachinePrecision] + t), $MachinePrecision] - N[(N[(n * N[(N[(t$95$1 * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 27.4%

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

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. count-2 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.4%

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

   9. Applied rewrites 53.4%

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

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

   1. Initial program 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. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.5%

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

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell}{Om \cdot Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \frac{\ell \cdot \ell}{Om}\\ t_4 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_3\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_5 := \frac{\ell}{Om} \cdot n\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\left(U - U*\right) \cdot n, t\_1, \frac{\ell + \ell}{Om}\right) \cdot \ell\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot \frac{\ell}{Om}, t\_5, \mathsf{fma}\left(-2, t\_3, t\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \left(\ell \cdot t\_1\right) \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\mathsf{fma}\left(\ell + \ell, \ell, \left(\ell \cdot t\_5\right) \cdot \left(U - U*\right)\right)}{Om}\right) \cdot \left(n + n\right)\right) \cdot U}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ l (* Om Om)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (/ (* l l) Om))
        (t_4
         (sqrt
          (* t_2 (- (- t (* 2.0 t_3)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_5 (* (/ l Om) n)))
   (if (<= t_4 0.0)
     (*
      (sqrt (* (- t (* (fma (* (- U U*) n) t_1 (/ (+ l l) Om)) l)) (+ n n)))
      (sqrt U))
     (if (<= t_4 2e+140)
       (sqrt (* t_2 (fma (* (- U* U) (/ l Om)) t_5 (fma -2.0 t_3 t))))
       (if (<= t_4 INFINITY)
         (sqrt
          (*
           (fma (- U* U) (* (* l t_1) n) (fma (* (/ l Om) l) -2.0 t))
           (* (+ n n) U)))
         (sqrt
          (*
           (* (- t (/ (fma (+ l l) l (* (* l t_5) (- U U*))) Om)) (+ n n))
           U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l / (Om * Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = (l * l) / Om;
	double t_4 = sqrt((t_2 * ((t - (2.0 * t_3)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_5 = (l / Om) * n;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((t - (fma(((U - U_42_) * n), t_1, ((l + l) / Om)) * l)) * (n + n))) * sqrt(U);
	} else if (t_4 <= 2e+140) {
		tmp = sqrt((t_2 * fma(((U_42_ - U) * (l / Om)), t_5, fma(-2.0, t_3, t))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((fma((U_42_ - U), ((l * t_1) * n), fma(((l / Om) * l), -2.0, t)) * ((n + n) * U)));
	} else {
		tmp = sqrt((((t - (fma((l + l), l, ((l * t_5) * (U - U_42_))) / Om)) * (n + n)) * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Float64(Om * Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(l * l) / Om)
	t_4 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_3)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_5 = Float64(Float64(l / Om) * n)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t - Float64(fma(Float64(Float64(U - U_42_) * n), t_1, Float64(Float64(l + l) / Om)) * l)) * Float64(n + n))) * sqrt(U));
	elseif (t_4 <= 2e+140)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(U_42_ - U) * Float64(l / Om)), t_5, fma(-2.0, t_3, t))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(fma(Float64(U_42_ - U), Float64(Float64(l * t_1) * n), fma(Float64(Float64(l / Om) * l), -2.0, t)) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(t - Float64(fma(Float64(l + l), l, Float64(Float64(l * t_5) * Float64(U - U_42_))) / Om)) * Float64(n + n)) * U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l / N[(Om * Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$3), $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$5 = N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * t$95$1 + N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+140], N[Sqrt[N[(t$95$2 * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * t$95$5 + N[(-2.0 * t$95$3 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l * t$95$1), $MachinePrecision] * n), $MachinePrecision] + N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t - N[(N[(N[(l + l), $MachinePrecision] * l + N[(N[(l * t$95$5), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $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*))))) < 0.0

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 27.4%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000012e140

   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

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 52.0%

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

   if 2.00000000000000012e140 < (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 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. count-2 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 49.7%

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

   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 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 55.7%

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

Alternative 4: 61.6% accurate, 0.3× speedup

Math

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

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*)))))))
   (if (<= t_1 0.0)
     (*
      (sqrt
       (*
        (- t (* (fma (* (- U U*) n) (/ l (* Om Om)) (/ (+ l l) Om)) l))
        (+ n n)))
      (sqrt U))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (fma (/ (* -2.0 l) Om) l t)
          (* (* n (/ (* (/ l Om) l) Om)) (- U U*)))))
       (sqrt
        (*
         (*
          (- t (/ (fma (+ l l) l (* (* l (* (/ l Om) n)) (- U U*))) Om))
          (+ n n))
         U))))))

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 tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((t - (fma(((U - U_42_) * n), (l / (Om * Om)), ((l + l) / Om)) * l)) * (n + n))) * sqrt(U);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((n + n) * U) * (fma(((-2.0 * l) / Om), l, t) - ((n * (((l / Om) * l) / Om)) * (U - U_42_)))));
	} else {
		tmp = sqrt((((t - (fma((l + l), l, ((l * ((l / Om) * n)) * (U - U_42_))) / Om)) * (n + n)) * U));
	}
	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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t - Float64(fma(Float64(Float64(U - U_42_) * n), Float64(l / Float64(Om * Om)), Float64(Float64(l + l) / Om)) * l)) * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) - Float64(Float64(n * Float64(Float64(Float64(l / Om) * l) / Om)) * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(Float64(Float64(t - Float64(fma(Float64(l + l), l, Float64(Float64(l * Float64(Float64(l / Om) * n)) * Float64(U - U_42_))) / Om)) * Float64(n + n)) * U));
	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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(l / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] - N[(N[(n * N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t - N[(N[(N[(l + l), $MachinePrecision] * l + N[(N[(l * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 27.4%

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

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. count-2 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.4%

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

   9. Applied rewrites 53.4%

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

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 55.7%

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

Alternative 5: 58.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell}{Om \cdot Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_3 := \sqrt{\left(\left(t - \frac{\mathsf{fma}\left(\ell + \ell, \ell, \left(\ell \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)}{Om}\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\left(U - U*\right) \cdot n, t\_1, \frac{\ell + \ell}{Om}\right) \cdot \ell\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+48}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \left(\ell \cdot t\_1\right) \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ l (* Om Om)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_3
         (sqrt
          (*
           (*
            (- t (/ (fma (+ l l) l (* (* l (* (/ l Om) n)) (- U U*))) Om))
            (+ n n))
           U))))
   (if (<= t_2 4e-153)
     (*
      (sqrt (* (- t (* (fma (* (- U U*) n) t_1 (/ (+ l l) Om)) l)) (+ n n)))
      (sqrt U))
     (if (<= t_2 5e+48)
       t_3
       (if (<= t_2 INFINITY)
         (sqrt
          (*
           (fma (- U* U) (* (* l t_1) n) (fma (* (/ l Om) l) -2.0 t))
           (* (+ n n) U)))
         t_3)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l / (Om * Om);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_3 = sqrt((((t - (fma((l + l), l, ((l * ((l / Om) * n)) * (U - U_42_))) / Om)) * (n + n)) * U));
	double tmp;
	if (t_2 <= 4e-153) {
		tmp = sqrt(((t - (fma(((U - U_42_) * n), t_1, ((l + l) / Om)) * l)) * (n + n))) * sqrt(U);
	} else if (t_2 <= 5e+48) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma((U_42_ - U), ((l * t_1) * n), fma(((l / Om) * l), -2.0, t)) * ((n + n) * U)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Float64(Om * Om))
	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_)))))
	t_3 = sqrt(Float64(Float64(Float64(t - Float64(fma(Float64(l + l), l, Float64(Float64(l * Float64(Float64(l / Om) * n)) * Float64(U - U_42_))) / Om)) * Float64(n + n)) * U))
	tmp = 0.0
	if (t_2 <= 4e-153)
		tmp = Float64(sqrt(Float64(Float64(t - Float64(fma(Float64(Float64(U - U_42_) * n), t_1, Float64(Float64(l + l) / Om)) * l)) * Float64(n + n))) * sqrt(U));
	elseif (t_2 <= 5e+48)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(fma(Float64(U_42_ - U), Float64(Float64(l * t_1) * n), fma(Float64(Float64(l / Om) * l), -2.0, t)) * Float64(Float64(n + n) * U)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l / N[(Om * Om), $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]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(t - N[(N[(N[(l + l), $MachinePrecision] * l + N[(N[(l * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 4e-153], N[(N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * t$95$1 + N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+48], t$95$3, If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l * t$95$1), $MachinePrecision] * n), $MachinePrecision] + N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

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*))))) < 4.00000000000000016e-153

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 27.4%

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

   if 4.00000000000000016e-153 < (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*))))) < 4.99999999999999973e48 or +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 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 55.7%

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

   if 4.99999999999999973e48 < (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 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. count-2 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 49.7%

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

Alternative 6: 58.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\mathsf{fma}\left(\left|\ell\right| + \left|\ell\right|, \left|\ell\right|, \left(\left|\ell\right| \cdot \left(\frac{\left|\ell\right|}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)}{Om}\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left|\ell\right| \cdot \mathsf{fma}\left(\frac{2}{Om}, \left|\ell\right|, \left(\frac{\left|\ell\right|}{Om \cdot Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= (fabs l) 2.15e+105)
   (sqrt
    (*
     (*
      (-
       t
       (/
        (fma
         (+ (fabs l) (fabs l))
         (fabs l)
         (* (* (fabs l) (* (/ (fabs l) Om) n)) (- U U*)))
        Om))
      (+ n n))
     U))
   (sqrt
    (*
     (* (* 2.0 n) U)
     (-
      t
      (*
       (fabs l)
       (fma
        (/ 2.0 Om)
        (fabs l)
        (* (* (/ (fabs l) (* Om Om)) n) (- U U*)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (fabs(l) <= 2.15e+105) {
		tmp = sqrt((((t - (fma((fabs(l) + fabs(l)), fabs(l), ((fabs(l) * ((fabs(l) / Om) * n)) * (U - U_42_))) / Om)) * (n + n)) * U));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (t - (fabs(l) * fma((2.0 / Om), fabs(l), (((fabs(l) / (Om * Om)) * n) * (U - U_42_)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (abs(l) <= 2.15e+105)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(fma(Float64(abs(l) + abs(l)), abs(l), Float64(Float64(abs(l) * Float64(Float64(abs(l) / Om) * n)) * Float64(U - U_42_))) / Om)) * Float64(n + n)) * U));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(abs(l) * fma(Float64(2.0 / Om), abs(l), Float64(Float64(Float64(abs(l) / Float64(Om * Om)) * n) * Float64(U - U_42_)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Abs[l], $MachinePrecision], 2.15e+105], N[Sqrt[N[(N[(N[(t - N[(N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[Abs[l], $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] * N[Abs[l], $MachinePrecision] + N[(N[(N[(N[Abs[l], $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1500000000000001e105

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 55.7%

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

   if 2.1500000000000001e105 < l

   1. Initial program 50.2%

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

   3. Applied rewrites 46.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 54.1%

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

Alternative 7: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right|}{Om}\\ \mathbf{if}\;\left|\ell\right| \leq 1.95 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\mathsf{fma}\left(\left|\ell\right| + \left|\ell\right|, \left|\ell\right|, \left(\left|\ell\right| \cdot \left(t\_1 \cdot n\right)\right) \cdot \left(U - U*\right)\right)}{Om}\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(-2, t\_1 \cdot \left|\ell\right|, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (fabs l) Om)))
   (if (<= (fabs l) 1.95e+105)
     (sqrt
      (*
       (*
        (-
         t
         (/
          (fma
           (+ (fabs l) (fabs l))
           (fabs l)
           (* (* (fabs l) (* t_1 n)) (- U U*)))
          Om))
        (+ n n))
       U))
     (sqrt (fabs (* (fma -2.0 (* t_1 (fabs l)) t) (* (+ n n) U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(l) / Om;
	double tmp;
	if (fabs(l) <= 1.95e+105) {
		tmp = sqrt((((t - (fma((fabs(l) + fabs(l)), fabs(l), ((fabs(l) * (t_1 * n)) * (U - U_42_))) / Om)) * (n + n)) * U));
	} else {
		tmp = sqrt(fabs((fma(-2.0, (t_1 * fabs(l)), t) * ((n + n) * U))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(abs(l) / Om)
	tmp = 0.0
	if (abs(l) <= 1.95e+105)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(fma(Float64(abs(l) + abs(l)), abs(l), Float64(Float64(abs(l) * Float64(t_1 * n)) * Float64(U - U_42_))) / Om)) * Float64(n + n)) * U));
	else
		tmp = sqrt(abs(Float64(fma(-2.0, Float64(t_1 * abs(l)), t) * Float64(Float64(n + n) * U))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 1.95e+105], N[Sqrt[N[(N[(N[(t - N[(N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(-2.0 * N[(t$95$1 * N[Abs[l], $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.94999999999999989e105

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 55.7%

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

   if 1.94999999999999989e105 < l

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   8. Applied rewrites 52.8%

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

Alternative 8: 57.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left|\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\mathsf{fma}\left(2, \ell, \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om} \cdot \ell\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (fabs (* (fma -2.0 (* (/ l Om) l) t) (* (+ n n) U))))))
   (if (<= n -5.2e+90)
     t_1
     (if (<= n 5.2e+77)
       (sqrt
        (*
         (* (- t (* (/ (fma 2.0 l (/ (* l (* n (- U U*))) Om)) Om) l)) (+ n n))
         U))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(fabs((fma(-2.0, ((l / Om) * l), t) * ((n + n) * U))));
	double tmp;
	if (n <= -5.2e+90) {
		tmp = t_1;
	} else if (n <= 5.2e+77) {
		tmp = sqrt((((t - ((fma(2.0, l, ((l * (n * (U - U_42_))) / Om)) / Om) * l)) * (n + n)) * U));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(abs(Float64(fma(-2.0, Float64(Float64(l / Om) * l), t) * Float64(Float64(n + n) * U))))
	tmp = 0.0
	if (n <= -5.2e+90)
		tmp = t_1;
	elseif (n <= 5.2e+77)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(fma(2.0, l, Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om)) / Om) * l)) * Float64(n + n)) * U));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[Abs[N[(N[(-2.0 * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -5.2e+90], t$95$1, If[LessEqual[n, 5.2e+77], N[Sqrt[N[(N[(N[(t - N[(N[(N[(2.0 * l + N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -5.1999999999999997e90 or 5.2000000000000004e77 < n

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   8. Applied rewrites 52.8%

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

   if -5.1999999999999997e90 < n < 5.2000000000000004e77

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 48.5%

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 53.4%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 54.4%

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

   10. Applied rewrites 54.4%

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

Alternative 9: 56.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \frac{\ell}{Om} \cdot \ell\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_1 \leq 10^{+146}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(t\_2, -2, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \]

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 (* (/ l Om) l)))
   (if (<= t_1 0.0)
     (* (sqrt (+ n n)) (sqrt (* U t)))
     (if (<= t_1 1e+146)
       (sqrt (* (fma -2.0 t_2 t) (* (+ n n) U)))
       (sqrt (fabs (* (* (fma t_2 -2.0 t) U) (+ n n))))))))

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 = (l / Om) * l;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * t));
	} else if (t_1 <= 1e+146) {
		tmp = sqrt((fma(-2.0, t_2, t) * ((n + n) * U)));
	} else {
		tmp = sqrt(fabs(((fma(t_2, -2.0, t) * U) * (n + n))));
	}
	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 = Float64(Float64(l / Om) * l)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	elseif (t_1 <= 1e+146)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(abs(Float64(Float64(fma(t_2, -2.0, t) * U) * Float64(n + n))));
	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[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+146], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(t$95$2 * -2.0 + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.8%

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

   3. Applied rewrites 53.8%

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 20.6%

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

   8. Applied rewrites 20.6%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.99999999999999934e145

   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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \color{blue}{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \color{blue}{U} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + \color{blue}{U} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   8. Applied rewrites 47.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   if 9.99999999999999934e145 < (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 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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \color{blue}{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \color{blue}{U} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + \color{blue}{U} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   8. Applied rewrites 47.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \cdot \sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}}} \]

   10. Applied rewrites 53.4%

       \[\leadsto \sqrt{\color{blue}{\left|\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (* (sqrt (+ n n)) (sqrt (* (- t (* l (* 2.0 (/ l Om)))) U)))
   (sqrt (fabs (* (fma -2.0 (* (/ l Om) l) t) (* (+ n n) U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt((n + n)) * sqrt(((t - (l * (2.0 * (l / Om)))) * U));
	} else {
		tmp = sqrt(fabs((fma(-2.0, ((l / Om) * l), t) * ((n + n) * U))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_))))) <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(Float64(t - Float64(l * Float64(2.0 * Float64(l / Om)))) * U)));
	else
		tmp = sqrt(abs(Float64(fma(-2.0, Float64(Float64(l / Om) * l), t) * Float64(Float64(n + n) * U))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(t - N[(l * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Abs[N[(N[(-2.0 * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot Om} \cdot n, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \color{blue}{\left(2 \cdot \frac{\ell}{Om}\right)}\right) \cdot U} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \left(2 \cdot \color{blue}{\frac{\ell}{Om}}\right)\right) \cdot U} \]

      2. lower-/.f64 27.5%

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{\color{blue}{Om}}\right)\right) \cdot U} \]

   8. Applied rewrites 27.5%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \color{blue}{\left(2 \cdot \frac{\ell}{Om}\right)}\right) \cdot U} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 52.8%

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (* (sqrt (+ n n)) (sqrt (* U t)))
   (sqrt (fabs (* (fma -2.0 (* (/ l Om) l) t) (* (+ n n) U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * t));
	} else {
		tmp = sqrt(fabs((fma(-2.0, ((l / Om) * l), t) * ((n + n) * U))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_))))) <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	else
		tmp = sqrt(abs(Float64(fma(-2.0, Float64(Float64(l / Om) * l), t) * Float64(Float64(n + n) * U))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Abs[N[(N[(-2.0 * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot Om} \cdot n, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   7. Step-by-step derivation

      1. lower-*.f64 20.6%

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{t}} \]

   8. Applied rewrites 20.6%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 52.8%

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 49.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (* (sqrt (+ n n)) (sqrt (* U t)))
   (sqrt (* (fma -2.0 (* (/ l Om) l) t) (* (+ n n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * t));
	} else {
		tmp = sqrt((fma(-2.0, ((l / Om) * l), t) * ((n + n) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_))))) <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	else
		tmp = sqrt(Float64(fma(-2.0, Float64(Float64(l / Om) * l), t) * Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-2.0 * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot Om} \cdot n, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   7. Step-by-step derivation

      1. lower-*.f64 20.6%

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{t}} \]

   8. Applied rewrites 20.6%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \color{blue}{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \color{blue}{U} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + \color{blue}{U} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   8. Applied rewrites 47.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.25e+89)
   (sqrt (* (+ U U) (* (fma -2.0 (* (/ l Om) l) t) n)))
   (sqrt (fabs (* (* (+ U U) t) n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.25e+89) {
		tmp = sqrt(((U + U) * (fma(-2.0, ((l / Om) * l), t) * n)));
	} else {
		tmp = sqrt(fabs((((U + U) * t) * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.25e+89)
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(-2.0, Float64(Float64(l / Om) * l), t) * n)));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.25e+89], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(-2.0 * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.24999999999999996e89

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. lower-*.f64 44.8%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \color{blue}{n}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \]

      11. pow2 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot n\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot n\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot n\right)} \]

      14. lower-*.f64 44.8%

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \color{blue}{n}\right)} \]

   8. Applied rewrites 48.1%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) \cdot n\right)}} \]

   if 1.24999999999999996e89 < t

   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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.7%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))
      INFINITY)
   (sqrt (* (* (+ U U) (fma -2.0 (* (/ l Om) l) t)) n))
   (sqrt (fabs (* (* (+ U U) t) n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= ((double) INFINITY)) {
		tmp = sqrt((((U + U) * fma(-2.0, ((l / Om) * l), t)) * n));
	} else {
		tmp = sqrt(fabs((((U + U) * t) * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_)))) <= Inf)
		tmp = sqrt(Float64(Float64(Float64(U + U) * fma(-2.0, Float64(Float64(l / Om) * l), t)) * n));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], Infinity], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * N[(-2.0 * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 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*)))) < +inf.0

   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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \color{blue}{n}\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \]

      10. pow2 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot n\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot n\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot n\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) \cdot \color{blue}{n}} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) \cdot \color{blue}{n}} \]

   8. Applied rewrites 47.7%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right) \cdot \color{blue}{n}} \]

   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 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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.7%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 40.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_1 \leq 10^{+292}:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (* (sqrt (+ n n)) (sqrt (* U t)))
     (if (<= t_1 1e+292)
       (sqrt (* (* (+ U U) n) t))
       (sqrt (fabs (* (* (+ U U) t) n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * t));
	} else if (t_1 <= 1e+292) {
		tmp = sqrt((((U + U) * n) * t));
	} else {
		tmp = sqrt(fabs((((U + U) * t) * 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) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((n + n)) * sqrt((u * t))
    else if (t_1 <= 1d+292) then
        tmp = sqrt((((u + u) * n) * t))
    else
        tmp = sqrt(abs((((u + u) * t) * 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 t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((n + n)) * Math.sqrt((U * t));
	} else if (t_1 <= 1e+292) {
		tmp = Math.sqrt((((U + U) * n) * t));
	} else {
		tmp = Math.sqrt(Math.abs((((U + U) * t) * n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((n + n)) * math.sqrt((U * t))
	elif t_1 <= 1e+292:
		tmp = math.sqrt((((U + U) * n) * t))
	else:
		tmp = math.sqrt(math.fabs((((U + U) * t) * n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	elseif (t_1 <= 1e+292)
		tmp = sqrt(Float64(Float64(Float64(U + U) * n) * t));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((n + n)) * sqrt((U * t));
	elseif (t_1 <= 1e+292)
		tmp = sqrt((((U + U) * n) * t));
	else
		tmp = sqrt(abs((((U + U) * t) * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+292], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.8%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot Om} \cdot n, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   7. Step-by-step derivation

      1. lower-*.f64 20.6%

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{t}} \]

   8. Applied rewrites 20.6%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e292

   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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      12. lower-*.f64 35.7%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      18. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      19. count-2-rev N/A

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      20. lower-+.f64 35.7%

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

   6. Applied rewrites 35.7%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

   if 1e292 < (*.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 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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.7%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 40.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \end{array} \]

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*)))))))
   (if (<= t_1 0.0)
     (sqrt (* (+ U U) (* t n)))
     (if (<= t_1 1e+146)
       (sqrt (* (* (+ U U) n) t))
       (sqrt (fabs (* (* (+ U U) t) n)))))))

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 tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((U + U) * (t * n)));
	} else if (t_1 <= 1e+146) {
		tmp = sqrt((((U + U) * n) * t));
	} else {
		tmp = sqrt(fabs((((U + U) * t) * 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((u + u) * (t * n)))
    else if (t_1 <= 1d+146) then
        tmp = sqrt((((u + u) * n) * t))
    else
        tmp = sqrt(abs((((u + u) * t) * 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 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 tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else if (t_1 <= 1e+146) {
		tmp = Math.sqrt((((U + U) * n) * t));
	} else {
		tmp = Math.sqrt(Math.abs((((U + U) * t) * n)));
	}
	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_)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((U + U) * (t * n)))
	elif t_1 <= 1e+146:
		tmp = math.sqrt((((U + U) * n) * t))
	else:
		tmp = math.sqrt(math.fabs((((U + U) * t) * n)))
	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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	elseif (t_1 <= 1e+146)
		tmp = sqrt(Float64(Float64(Float64(U + U) * n) * t));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	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_)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((U + U) * (t * n)));
	elseif (t_1 <= 1e+146)
		tmp = sqrt((((U + U) * n) * t));
	else
		tmp = sqrt(abs((((U + U) * t) * n)));
	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]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+146], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-+.f64 35.9%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 35.9%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.99999999999999934e145

   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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      12. lower-*.f64 35.7%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      18. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      19. count-2-rev N/A

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      20. lower-+.f64 35.7%

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

   6. Applied rewrites 35.7%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

   if 9.99999999999999934e145 < (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 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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.7%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 38.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))
      0.0)
   (sqrt (* (* (+ U U) t) n))
   (sqrt (* (* (+ U U) n) t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt((((U + U) * t) * n));
	} else {
		tmp = sqrt((((U + U) * n) * t));
	}
	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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))) <= 0.0d0) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt((((u + u) * n) * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt((((U + U) * n) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 0.0:
		tmp = math.sqrt((((U + U) * t) * n))
	else:
		tmp = math.sqrt((((U + U) * n) * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_)))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(U + U) * t) * n));
	else
		tmp = sqrt(Float64(Float64(Float64(U + U) * n) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))) <= 0.0)
		tmp = sqrt((((U + U) * t) * n));
	else
		tmp = sqrt((((U + U) * n) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

      10. lower-+.f64 35.6%

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

   6. Applied rewrites 35.6%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   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. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. 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 35.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      12. lower-*.f64 35.7%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      18. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      19. count-2-rev N/A

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      20. lower-+.f64 35.7%

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

   6. Applied rewrites 35.7%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.7% accurate, 4.7× speedup

Math

\[\sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* (+ U U) n) t)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((U + U) * n) * 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((((u + u) * n) * t))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((U + U) * n) * t));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((U + U) * n) * t))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(U + U) * n) * t))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((U + U) * n) * t));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

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. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
3. 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 35.9%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 35.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   9. associate-*l* N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   12. lower-*.f64 35.7%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   15. associate-*l* N/A

       \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   16. *-commutative N/A

       \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

   17. associate-*r* N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

   18. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

   19. count-2-rev N/A

       \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

   20. lower-+.f64 35.7%

       \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

6. Applied rewrites 35.7%

   \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025179 
(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*))))))