Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 64.3%

Time: 7.7s

Alternatives: 9

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 64.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(2, \ell, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;U \leq -1000000:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t\_1}\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(t\_1 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ l Om) (fma 2.0 l (* (* (/ l Om) n) (- U U*)))))))
   (if (<= U -1000000.0)
     (sqrt (* (* U (+ n n)) t_1))
     (if (<= U 1.4e+98)
       (sqrt (* (+ n n) (* t_1 U)))
       (* (sqrt (* (fma (* l (/ l Om)) -2.0 t) (+ n n))) (sqrt U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * fma(2.0, l, (((l / Om) * n) * (U - U_42_))));
	double tmp;
	if (U <= -1000000.0) {
		tmp = sqrt(((U * (n + n)) * t_1));
	} else if (U <= 1.4e+98) {
		tmp = sqrt(((n + n) * (t_1 * U)));
	} else {
		tmp = sqrt((fma((l * (l / Om)), -2.0, t) * (n + n))) * sqrt(U);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l / Om) * fma(2.0, l, Float64(Float64(Float64(l / Om) * n) * Float64(U - U_42_)))))
	tmp = 0.0
	if (U <= -1000000.0)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t_1));
	elseif (U <= 1.4e+98)
		tmp = sqrt(Float64(Float64(n + n) * Float64(t_1 * U)));
	else
		tmp = Float64(sqrt(Float64(fma(Float64(l * Float64(l / Om)), -2.0, t) * Float64(n + n))) * sqrt(U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l + N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1000000.0], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 1.4e+98], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(t$95$1 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if U < -1e6

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 56.1

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate--l- N/A

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

   7. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 57.5

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 60.2%

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

   if -1e6 < U < 1.4e98

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 56.1

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate--l- N/A

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

   7. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.8

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

   9. Applied rewrites 60.5%

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

   if 1.4e98 < U

   1. Initial program 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.2%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 28.8%

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

Alternative 2: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(2, \ell, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;n \leq 2.7 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1 \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ l Om) (fma 2.0 l (* (* (/ l Om) n) (- U U*)))))))
   (if (<= n 2.7e-306)
     (sqrt (* (* U (+ n n)) t_1))
     (* (sqrt (+ n n)) (sqrt (* t_1 U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l / Om) * fma(2.0, l, (((l / Om) * n) * (U - U_42_))));
	double tmp;
	if (n <= 2.7e-306) {
		tmp = sqrt(((U * (n + n)) * t_1));
	} else {
		tmp = sqrt((n + n)) * sqrt((t_1 * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l / Om) * fma(2.0, l, Float64(Float64(Float64(l / Om) * n) * Float64(U - U_42_)))))
	tmp = 0.0
	if (n <= 2.7e-306)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t_1));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(t_1 * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l + N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 2.7e-306], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 2.70000000000000009e-306

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 56.1

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate--l- N/A

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

   7. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 57.5

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 60.2%

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

   if 2.70000000000000009e-306 < n

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 56.1

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate--l- N/A

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

   7. Applied rewrites 57.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 32.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 32.6

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

   9. Applied rewrites 34.0%

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

Alternative 3: 63.3% accurate, 0.6× speedup

Math

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

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) (fma (* l (/ l Om)) -2.0 t)) U))
   (sqrt
    (*
     (* U (+ n n))
     (- t (* (/ l Om) (fma 2.0 l (* (* (/ l Om) n) (- U 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) * fma((l * (l / Om)), -2.0, t)) * U));
	} else {
		tmp = sqrt(((U * (n + n)) * (t - ((l / Om) * fma(2.0, l, (((l / Om) * n) * (U - U_42_)))))));
	}
	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 = sqrt(Float64(Float64(Float64(n + n) * fma(Float64(l * Float64(l / Om)), -2.0, t)) * U));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * Float64(t - Float64(Float64(l / Om) * fma(2.0, l, Float64(Float64(Float64(l / Om) * n) * Float64(U - U_42_)))))));
	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[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l + N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.2%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 44.6

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 44.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 48.5%

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

   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 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 56.1

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

      9. lift--.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate--l- N/A

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

   7. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 57.5

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 60.2%

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

Alternative 4: 52.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\\ \mathbf{if}\;U \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot t\_1\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (* l (/ l Om)) -2.0 t)))
   (if (<= U -1e-309)
     (sqrt (* (* (+ n n) t_1) U))
     (* (sqrt (* t_1 (+ n n))) (sqrt U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((l * (l / Om)), -2.0, t);
	double tmp;
	if (U <= -1e-309) {
		tmp = sqrt((((n + n) * t_1) * U));
	} else {
		tmp = sqrt((t_1 * (n + n))) * sqrt(U);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(l * Float64(l / Om)), -2.0, t)
	tmp = 0.0
	if (U <= -1e-309)
		tmp = sqrt(Float64(Float64(Float64(n + n) * t_1) * U));
	else
		tmp = Float64(sqrt(Float64(t_1 * Float64(n + n))) * sqrt(U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[U, -1e-309], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t$95$1 * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -1.000000000000002e-309

   1. Initial program 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.2%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 44.6

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 44.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 48.5%

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

   if -1.000000000000002e-309 < U

   1. Initial program 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.2%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 28.8%

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

Alternative 5: 50.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 44.2%

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

4. Taylor expanded in n around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 44.6

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

6. Applied rewrites 44.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. count-2-rev N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 44.6

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

   7. lift-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lift-+.f64 44.6

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-pow.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

8. Applied rewrites 48.5%

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

Alternative 6: 48.5% accurate, 0.7× speedup

Math

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

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 (* U (* (+ n n) t)))
   (sqrt (* (fma (* l (/ l Om)) -2.0 t) (* U (+ n n))))))

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((U * ((n + n) * t)));
	} else {
		tmp = sqrt((fma((l * (l / Om)), -2.0, t) * (U * (n + n))));
	}
	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 = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	else
		tmp = sqrt(Float64(fma(Float64(l * Float64(l / Om)), -2.0, t) * Float64(U * Float64(n + n))));
	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[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(U * N[(n + n), $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 50.3%

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

   2. 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 36.0

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.7

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

   6. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.7

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.7

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

   8. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.0

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

   10. Applied rewrites 36.0%

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

   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*)))))

   1. Initial program 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.2%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 44.5

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

   8. Applied rewrites 48.1%

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

Alternative 7: 39.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.6e-306)
   (sqrt (* U (* (+ n n) t)))
   (* t (* (sqrt (* 2.0 (* U n))) (sqrt (/ 1.0 t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.6e-306) {
		tmp = sqrt((U * ((n + n) * t)));
	} else {
		tmp = t * (sqrt((2.0 * (U * n))) * sqrt((1.0 / 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 (t <= 1.6d-306) then
        tmp = sqrt((u * ((n + n) * t)))
    else
        tmp = t * (sqrt((2.0d0 * (u * n))) * sqrt((1.0d0 / 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 (t <= 1.6e-306) {
		tmp = Math.sqrt((U * ((n + n) * t)));
	} else {
		tmp = t * (Math.sqrt((2.0 * (U * n))) * Math.sqrt((1.0 / t)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.6e-306:
		tmp = math.sqrt((U * ((n + n) * t)))
	else:
		tmp = t * (math.sqrt((2.0 * (U * n))) * math.sqrt((1.0 / t)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.6e-306)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	else
		tmp = Float64(t * Float64(sqrt(Float64(2.0 * Float64(U * n))) * sqrt(Float64(1.0 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.6e-306)
		tmp = sqrt((U * ((n + n) * t)));
	else
		tmp = t * (sqrt((2.0 * (U * n))) * sqrt((1.0 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.6e-306], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t * N[(N[Sqrt[N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999985e-306

   1. Initial program 50.3%

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

   2. 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 36.0

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.7

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

   6. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.7

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.7

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

   8. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.0

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

   10. Applied rewrites 36.0%

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

   if 1.59999999999999985e-306 < t

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   7. Applied rewrites 33.6%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 21.5

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

   10. Applied rewrites 21.5%

       \[\leadsto \color{blue}{t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n + n\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.6e-306)
   (sqrt (* U (* (+ n n) t)))
   (* (sqrt (* U (+ n n))) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.6e-306) {
		tmp = sqrt((U * ((n + n) * t)));
	} else {
		tmp = sqrt((U * (n + n))) * sqrt(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 (t <= 1.6d-306) then
        tmp = sqrt((u * ((n + n) * t)))
    else
        tmp = sqrt((u * (n + n))) * sqrt(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 (t <= 1.6e-306) {
		tmp = Math.sqrt((U * ((n + n) * t)));
	} else {
		tmp = Math.sqrt((U * (n + n))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.6e-306:
		tmp = math.sqrt((U * ((n + n) * t)))
	else:
		tmp = math.sqrt((U * (n + n))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.6e-306)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n + n))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.6e-306)
		tmp = sqrt((U * ((n + n) * t)));
	else
		tmp = sqrt((U * (n + n))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.6e-306], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999985e-306

   1. Initial program 50.3%

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

   2. 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 36.0

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

   4. Applied rewrites 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.7

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

   6. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.7

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.7

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

   8. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.0

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

   10. Applied rewrites 36.0%

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

   if 1.59999999999999985e-306 < t

   1. Initial program 50.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   7. Applied rewrites 33.6%

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

   8. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 21.5

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

   10. Applied rewrites 21.5%

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

Alternative 9: 36.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

2. 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 36.0

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

4. Applied rewrites 36.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 36.7

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

6. Applied rewrites 36.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 36.7

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 36.7

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

   13. lift-*.f64 N/A

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

   14. count-2-rev N/A

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

   15. lift-+.f64 36.7

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

8. Applied rewrites 36.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 36.0

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

10. Applied rewrites 36.0%

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

Reproduce

herbie shell --seed 2025148 
(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*))))))