Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 65.9%

Time: 11.2s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\\ t_2 := \frac{n \cdot \left|\ell\right|}{Om}\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \frac{\left|\ell\right|}{Om}\\ t_5 := t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {t\_4}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_6 := \left|\ell\right| + \left|\ell\right|\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\left(t - \frac{\mathsf{fma}\left(U - U*, t\_2, t\_6\right)}{Om} \cdot \left|\ell\right|\right) \cdot U\right)}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\sqrt{t\_3 \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot t\_4, t\_4 \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\sqrt{\left(t - \left|\ell\right| \cdot \mathsf{fma}\left(\frac{t\_2}{Om}, U - U*, \frac{t\_6}{Om}\right)\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left|\ell\right| \cdot \sqrt{-1 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* (fabs l) (fabs l)) Om))
        (t_2 (/ (* n (fabs l)) Om))
        (t_3 (* (* 2.0 n) U))
        (t_4 (/ (fabs l) Om))
        (t_5 (* t_3 (- (- t (* 2.0 t_1)) (* (* n (pow t_4 2.0)) (- U U*)))))
        (t_6 (+ (fabs l) (fabs l))))
   (if (<= t_5 0.0)
     (sqrt (* (+ n n) (* (- t (* (/ (fma (- U U*) t_2 t_6) Om) (fabs l))) U)))
     (if (<= t_5 5e+246)
       (sqrt (* t_3 (fma (* (- U* U) t_4) (* t_4 n) (fma -2.0 t_1 t))))
       (if (<= t_5 INFINITY)
         (sqrt
          (*
           (- t (* (fabs l) (fma (/ t_2 Om) (- U U*) (/ t_6 Om))))
           (* U (+ n n))))
         (*
          (sqrt 2.0)
          (*
           (fabs l)
           (sqrt
            (*
             -1.0
             (*
              U
              (*
               n
               (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (fabs(l) * fabs(l)) / Om;
	double t_2 = (n * fabs(l)) / Om;
	double t_3 = (2.0 * n) * U;
	double t_4 = fabs(l) / Om;
	double t_5 = t_3 * ((t - (2.0 * t_1)) - ((n * pow(t_4, 2.0)) * (U - U_42_)));
	double t_6 = fabs(l) + fabs(l);
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt(((n + n) * ((t - ((fma((U - U_42_), t_2, t_6) / Om) * fabs(l))) * U)));
	} else if (t_5 <= 5e+246) {
		tmp = sqrt((t_3 * fma(((U_42_ - U) * t_4), (t_4 * n), fma(-2.0, t_1, t))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt(((t - (fabs(l) * fma((t_2 / Om), (U - U_42_), (t_6 / Om)))) * (U * (n + n))));
	} else {
		tmp = sqrt(2.0) * (fabs(l) * sqrt((-1.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(abs(l) * abs(l)) / Om)
	t_2 = Float64(Float64(n * abs(l)) / Om)
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(abs(l) / Om)
	t_5 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (t_4 ^ 2.0)) * Float64(U - U_42_))))
	t_6 = Float64(abs(l) + abs(l))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = sqrt(Float64(Float64(n + n) * Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), t_2, t_6) / Om) * abs(l))) * U)));
	elseif (t_5 <= 5e+246)
		tmp = sqrt(Float64(t_3 * fma(Float64(Float64(U_42_ - U) * t_4), Float64(t_4 * n), fma(-2.0, t_1, t))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(Float64(t - Float64(abs(l) * fma(Float64(t_2 / Om), Float64(U - U_42_), Float64(t_6 / Om)))) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(sqrt(2.0) * Float64(abs(l) * sqrt(Float64(-1.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0)))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * t$95$2 + t$95$6), $MachinePrecision] / Om), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 5e+246], N[Sqrt[N[(t$95$3 * N[(N[(N[(U$42$ - U), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(N[(t - N[(N[Abs[l], $MachinePrecision] * N[(N[(t$95$2 / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(t$95$6 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-1.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.99999999999999976e246

   1. Initial program 49.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 50.8%

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

   if 4.99999999999999976e246 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 2: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\\ t_2 := \frac{n \cdot \left|\ell\right|}{Om}\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \frac{\left|\ell\right|}{Om}\\ t_5 := t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {t\_4}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_6 := \left|\ell\right| + \left|\ell\right|\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\left(t - \frac{\mathsf{fma}\left(U - U*, t\_2, t\_6\right)}{Om} \cdot \left|\ell\right|\right) \cdot U\right)}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\sqrt{t\_3 \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot t\_4, t\_4 \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\sqrt{\left(t - \left|\ell\right| \cdot \mathsf{fma}\left(\frac{t\_2}{Om}, U - U*, \frac{t\_6}{Om}\right)\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* (fabs l) (fabs l)) Om))
        (t_2 (/ (* n (fabs l)) Om))
        (t_3 (* (* 2.0 n) U))
        (t_4 (/ (fabs l) Om))
        (t_5 (* t_3 (- (- t (* 2.0 t_1)) (* (* n (pow t_4 2.0)) (- U U*)))))
        (t_6 (+ (fabs l) (fabs l))))
   (if (<= t_5 0.0)
     (sqrt (* (+ n n) (* (- t (* (/ (fma (- U U*) t_2 t_6) Om) (fabs l))) U)))
     (if (<= t_5 5e+246)
       (sqrt (* t_3 (fma (* (- U* U) t_4) (* t_4 n) (fma -2.0 t_1 t))))
       (if (<= t_5 INFINITY)
         (sqrt
          (*
           (- t (* (fabs l) (fma (/ t_2 Om) (- U U*) (/ t_6 Om))))
           (* U (+ n n))))
         (*
          (fabs l)
          (sqrt
           (*
            -2.0
            (*
             U
             (*
              n
              (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (fabs(l) * fabs(l)) / Om;
	double t_2 = (n * fabs(l)) / Om;
	double t_3 = (2.0 * n) * U;
	double t_4 = fabs(l) / Om;
	double t_5 = t_3 * ((t - (2.0 * t_1)) - ((n * pow(t_4, 2.0)) * (U - U_42_)));
	double t_6 = fabs(l) + fabs(l);
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt(((n + n) * ((t - ((fma((U - U_42_), t_2, t_6) / Om) * fabs(l))) * U)));
	} else if (t_5 <= 5e+246) {
		tmp = sqrt((t_3 * fma(((U_42_ - U) * t_4), (t_4 * n), fma(-2.0, t_1, t))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt(((t - (fabs(l) * fma((t_2 / Om), (U - U_42_), (t_6 / Om)))) * (U * (n + n))));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(abs(l) * abs(l)) / Om)
	t_2 = Float64(Float64(n * abs(l)) / Om)
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(abs(l) / Om)
	t_5 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (t_4 ^ 2.0)) * Float64(U - U_42_))))
	t_6 = Float64(abs(l) + abs(l))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = sqrt(Float64(Float64(n + n) * Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), t_2, t_6) / Om) * abs(l))) * U)));
	elseif (t_5 <= 5e+246)
		tmp = sqrt(Float64(t_3 * fma(Float64(Float64(U_42_ - U) * t_4), Float64(t_4 * n), fma(-2.0, t_1, t))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(Float64(t - Float64(abs(l) * fma(Float64(t_2 / Om), Float64(U - U_42_), Float64(t_6 / Om)))) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * t$95$2 + t$95$6), $MachinePrecision] / Om), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 5e+246], N[Sqrt[N[(t$95$3 * N[(N[(N[(U$42$ - U), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(N[(t - N[(N[Abs[l], $MachinePrecision] * N[(N[(t$95$2 / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(t$95$6 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.99999999999999976e246

   1. Initial program 49.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 50.8%

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

   if 4.99999999999999976e246 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

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

   1. Initial program 49.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.9

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

   4. Applied rewrites 15.9%

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

Alternative 3: 65.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   if +inf.0 < (*.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 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 4: 65.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

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

   1. Initial program 49.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.7

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

   3. Applied rewrites 53.7%

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

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

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 5: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := t - \frac{\mathsf{fma}\left(U - U*, \frac{n \cdot \ell}{Om}, \ell + \ell\right)}{Om} \cdot \ell\\ \mathbf{if}\;U \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\ell}{Om \cdot Om} \cdot n, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(t\_1 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ (fma (- U U*) (/ (* n l) Om) (+ l l)) Om) l))))
   (if (<= U -1.1e+49)
     (sqrt
      (*
       (- t (* l (fma (* (/ l (* Om Om)) n) (- U U*) (/ (+ l l) Om))))
       (* U (+ n n))))
     (if (<= U 4.8e-308)
       (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 = t - ((fma((U - U_42_), ((n * l) / Om), (l + l)) / Om) * l);
	double tmp;
	if (U <= -1.1e+49) {
		tmp = sqrt(((t - (l * fma(((l / (Om * Om)) * n), (U - U_42_), ((l + l) / Om)))) * (U * (n + n))));
	} else if (U <= 4.8e-308) {
		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 = Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(Float64(n * l) / Om), Float64(l + l)) / Om) * l))
	tmp = 0.0
	if (U <= -1.1e+49)
		tmp = sqrt(Float64(Float64(t - Float64(l * fma(Float64(Float64(l / Float64(Om * Om)) * n), Float64(U - U_42_), Float64(Float64(l + l) / Om)))) * Float64(U * Float64(n + n))));
	elseif (U <= 4.8e-308)
		tmp = sqrt(Float64(Float64(n + n) * Float64(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[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.1e+49], N[Sqrt[N[(N[(t - N[(l * N[(N[(N[(l / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 4.8e-308], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(t$95$1 * U), $MachinePrecision]), $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 3 regimes

2. if U < -1.1e49

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

   if -1.1e49 < U < 4.80000000000000016e-308

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

   if 4.80000000000000016e-308 < U

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   9. Applied rewrites 34.1%

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

Alternative 6: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := t - \frac{\mathsf{fma}\left(U - U*, \frac{n \cdot \ell}{Om}, \ell + \ell\right)}{Om} \cdot \ell\\ \mathbf{if}\;U \leq -4.8 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left|\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(t\_1 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (/ (fma (- U U*) (/ (* n l) Om) (+ l l)) Om) l))))
   (if (<= U -4.8e+138)
     (sqrt (fabs (* (fma (/ (* -2.0 l) Om) l t) (* (+ n n) U))))
     (if (<= U 4.8e-308)
       (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 = t - ((fma((U - U_42_), ((n * l) / Om), (l + l)) / Om) * l);
	double tmp;
	if (U <= -4.8e+138) {
		tmp = sqrt(fabs((fma(((-2.0 * l) / Om), l, t) * ((n + n) * U))));
	} else if (U <= 4.8e-308) {
		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 = Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(Float64(n * l) / Om), Float64(l + l)) / Om) * l))
	tmp = 0.0
	if (U <= -4.8e+138)
		tmp = sqrt(abs(Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * Float64(Float64(n + n) * U))));
	elseif (U <= 4.8e-308)
		tmp = sqrt(Float64(Float64(n + n) * Float64(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[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -4.8e+138], N[Sqrt[N[Abs[N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 4.8e-308], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(t$95$1 * U), $MachinePrecision]), $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 3 regimes

2. if U < -4.8000000000000002e138

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.0

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

   6. Applied rewrites 44.0%

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

   8. Applied rewrites 53.3%

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

   if -4.8000000000000002e138 < U < 4.80000000000000016e-308

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

   if 4.80000000000000016e-308 < U

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   9. Applied rewrites 34.1%

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

Alternative 7: 60.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n \cdot \ell}{Om}, \ell + \ell\right)}{Om} \cdot \ell\right) \cdot U\\ \mathbf{if}\;n \leq -5 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (- t (* (/ (fma (- U U*) (/ (* n l) Om) (+ l l)) Om) l)) U)))
   (if (<= n -5e-203)
     (sqrt (* (+ n n) t_1))
     (if (<= n 1.08e-86)
       (sqrt (* (+ U U) (* (fma (/ (* -2.0 l) Om) l t) n)))
       (* (sqrt (+ n n)) (sqrt t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (t - ((fma((U - U_42_), ((n * l) / Om), (l + l)) / Om) * l)) * U;
	double tmp;
	if (n <= -5e-203) {
		tmp = sqrt(((n + n) * t_1));
	} else if (n <= 1.08e-86) {
		tmp = sqrt(((U + U) * (fma(((-2.0 * l) / Om), l, t) * n)));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(Float64(n * l) / Om), Float64(l + l)) / Om) * l)) * U)
	tmp = 0.0
	if (n <= -5e-203)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	elseif (n <= 1.08e-86)
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * n)));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, -5e-203], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.08e-86], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.0000000000000002e-203

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.7

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

   9. Applied rewrites 57.3%

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

   if -5.0000000000000002e-203 < n < 1.07999999999999993e-86

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.0

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

   6. Applied rewrites 44.0%

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

   8. Applied rewrites 48.2%

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

   if 1.07999999999999993e-86 < n

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. lower-*.f64 31.3

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

   9. Applied rewrites 33.2%

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

Alternative 8: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left|\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \mathbf{if}\;U \leq -4.8 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n \cdot \ell}{Om}, \ell + \ell\right)}{Om} \cdot \ell\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (fabs (* (fma (/ (* -2.0 l) Om) l t) (* (+ n n) U))))))
   (if (<= U -4.8e+138)
     t_1
     (if (<= U 2.7e+23)
       (sqrt
        (*
         (+ n n)
         (* (- t (* (/ (fma (- U U*) (/ (* n l) Om) (+ l l)) Om) l)) U)))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(fabs((fma(((-2.0 * l) / Om), l, t) * ((n + n) * U))));
	double tmp;
	if (U <= -4.8e+138) {
		tmp = t_1;
	} else if (U <= 2.7e+23) {
		tmp = sqrt(((n + n) * ((t - ((fma((U - U_42_), ((n * l) / Om), (l + l)) / Om) * l)) * U)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(abs(Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * Float64(Float64(n + n) * U))))
	tmp = 0.0
	if (U <= -4.8e+138)
		tmp = t_1;
	elseif (U <= 2.7e+23)
		tmp = sqrt(Float64(Float64(n + n) * Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(Float64(n * l) / Om), Float64(l + l)) / Om) * l)) * U)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[Abs[N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -4.8e+138], t$95$1, If[LessEqual[U, 2.7e+23], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] + N[(l + l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.8000000000000002e138 or 2.6999999999999999e23 < U

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.0

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

   6. Applied rewrites 44.0%

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

   8. Applied rewrites 53.3%

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

   if -4.8000000000000002e138 < U < 2.6999999999999999e23

   1. Initial program 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.7

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\frac{n \cdot \ell}{Om}}{Om}, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot \left(U \cdot \left(n + n\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(t - \ell \cdot \mathsf{fma}\left(\frac{\frac{n \cdot \ell}{Om}}{Om}, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot \color{blue}{\left(U \cdot \left(n + n\right)\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \ell \cdot \mathsf{fma}\left(\frac{\frac{n \cdot \ell}{Om}}{Om}, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U\right) \cdot \left(n + n\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\left(t - \ell \cdot \mathsf{fma}\left(\frac{\frac{n \cdot \ell}{Om}}{Om}, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\left(t - \ell \cdot \mathsf{fma}\left(\frac{\frac{n \cdot \ell}{Om}}{Om}, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U\right)}} \]

      6. lower-*.f64 53.7

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(t - \ell \cdot \mathsf{fma}\left(\frac{\frac{n \cdot \ell}{Om}}{Om}, U - U*, \frac{\ell + \ell}{Om}\right)\right) \cdot U\right)}} \]

   9. Applied rewrites 57.3%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n \cdot \ell}{Om}, \ell + \ell\right)}{Om} \cdot \ell\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right)\\ \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t\_1 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t\_1 \cdot \left(\left(n + n\right) \cdot U\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ (* -2.0 l) Om) l t)))
   (if (<=
        (sqrt
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        0.0)
     (sqrt (* (+ U U) (* t_1 n)))
     (sqrt (fabs (* t_1 (* (+ n n) U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(((-2.0 * l) / Om), l, t);
	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 + U) * (t_1 * n)));
	} else {
		tmp = sqrt(fabs((t_1 * ((n + n) * U))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(Float64(-2.0 * l) / Om), l, t)
	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(U + U) * Float64(t_1 * n)));
	else
		tmp = sqrt(abs(Float64(t_1 * Float64(Float64(n + n) * U))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision]}, 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[(U + U), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t$95$1 * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 48.2%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 53.3%

      \[\leadsto \sqrt{\color{blue}{\left|\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (+ U U) (* (fma (/ (* -2.0 l) Om) l t) n)))
     (if (<= t_1 5e+245)
       (sqrt (* (fma (* l (/ -2.0 Om)) l t) (* (+ n n) U)))
       (sqrt (fabs (* (* (fma (* -2.0 (/ l Om)) l t) U) (+ n n))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((U + U) * (fma(((-2.0 * l) / Om), l, t) * n)));
	} else if (t_1 <= 5e+245) {
		tmp = sqrt((fma((l * (-2.0 / Om)), l, t) * ((n + n) * U)));
	} else {
		tmp = sqrt(fabs(((fma((-2.0 * (l / Om)), l, t) * U) * (n + n))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * n)));
	elseif (t_1 <= 5e+245)
		tmp = sqrt(Float64(fma(Float64(l * Float64(-2.0 / Om)), l, t) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(abs(Float64(Float64(fma(Float64(-2.0 * Float64(l / Om)), l, t) * U) * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+245], N[Sqrt[N[(N[(N[(l * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 48.2%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000034e245

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 46.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(2 \cdot \ell\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      8. count-2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(2 \cdot \ell\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell \cdot -2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      14. lower-/.f64 46.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

   10. Applied rewrites 46.9%

       \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

   if 5.00000000000000034e245 < (*.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 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 46.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]

   10. Applied rewrites 53.6%

       \[\leadsto \sqrt{\color{blue}{\left|\left(\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(\sqrt{\left(-2 \cdot U\right) \cdot \left(U - U*\right)} \cdot \left|\ell\right|\right) \cdot \frac{n}{Om}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (+ U U) (* (fma (/ (* -2.0 l) Om) l t) n)))
     (if (<= t_1 INFINITY)
       (sqrt (* (fma (* l (/ -2.0 Om)) l t) (* (+ n n) U)))
       (- (* (* (sqrt (* (* -2.0 U) (- U U*))) (fabs l)) (/ n Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((U + U) * (fma(((-2.0 * l) / Om), l, t) * n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((fma((l * (-2.0 / Om)), l, t) * ((n + n) * U)));
	} else {
		tmp = -((sqrt(((-2.0 * U) * (U - U_42_))) * fabs(l)) * (n / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * n)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(fma(Float64(l * Float64(-2.0 / Om)), l, t) * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(-Float64(Float64(sqrt(Float64(Float64(-2.0 * U) * Float64(U - U_42_))) * abs(l)) * Float64(n / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(l * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[(N[(N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 48.2%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 46.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(2 \cdot \ell\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      8. count-2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(2 \cdot \ell\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell \cdot -2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      14. lower-/.f64 46.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

   10. Applied rewrites 46.9%

       \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around 0

      \[\leadsto \color{blue}{\frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{\color{blue}{Om}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      9. lower--.f64 10.6

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

   4. Applied rewrites 10.6%

      \[\leadsto \color{blue}{\frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}{Om} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{\left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      7. pow2 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\left({n}^{2} \cdot \left(U - U*\right)\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left({n}^{2} \cdot \left(U - U*\right)\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left({n}^{2} \cdot \left(U - U*\right)\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left({n}^{2} \cdot \left(U - U*\right)\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left({n}^{2} \cdot \left(U - U*\right)\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left({n}^{2} \cdot \left(U - U*\right)\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      16. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(U - U*\right) \cdot {n}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(U - U*\right) \cdot {n}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      18. unpow2 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      19. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(\left(U - U*\right) \cdot n\right) \cdot n\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(\left(U - U*\right) \cdot n\right) \cdot n\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(\left(U - U*\right) \cdot n\right) \cdot n\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

      22. lower-*.f64 12.3

          \[\leadsto \frac{\sqrt{\left(\left(\left(\left(\left(U - U*\right) \cdot n\right) \cdot n\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

   6. Applied rewrites 12.3%

      \[\leadsto \frac{\sqrt{\left(\left(\left(\left(\left(U - U*\right) \cdot n\right) \cdot n\right) \cdot \ell\right) \cdot \ell\right) \cdot \left(-2 \cdot U\right)}}{Om} \]

   7. Taylor expanded in n around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{\color{blue}{Om}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      7. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      8. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

      9. lower--.f64 11.8

         \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

   9. Applied rewrites 11.8%

      \[\leadsto -1 \cdot \color{blue}{\frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{\color{blue}{Om}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om}\right) \]

       3. lower-neg.f64 11.8

          \[\leadsto -\frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

       4. lift-/.f64 N/A

          \[\leadsto -\frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

       5. lift-*.f64 N/A

          \[\leadsto -\frac{n \cdot \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)}}{Om} \]

       6. *-commutative N/A

          \[\leadsto -\frac{\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot n}{Om} \]

       7. associate-/l* N/A

          \[\leadsto -\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \frac{n}{Om} \]

       8. lower-*.f64 N/A

          \[\leadsto -\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)\right)} \cdot \frac{n}{Om} \]

   11. Applied rewrites 14.1%

       \[\leadsto -\left(\sqrt{\left(-2 \cdot U\right) \cdot \left(U - U*\right)} \cdot \left|\ell\right|\right) \cdot \frac{n}{Om} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (sqrt (* (+ U U) (* (fma (/ (* -2.0 l) Om) l t) n)))
     (if (<= t_1 INFINITY)
       (sqrt (* (fma (* l (/ -2.0 Om)) l t) (* (+ n n) U)))
       (sqrt (fabs (* t (* U (+ n n)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((U + U) * (fma(((-2.0 * l) / Om), l, t) * n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((fma((l * (-2.0 / Om)), l, t) * ((n + n) * U)));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * n)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(fma(Float64(l * Float64(-2.0 / Om)), l, t) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(l * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 48.2%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 46.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(2 \cdot \ell\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      8. count-2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\mathsf{neg}\left(2 \cdot \ell\right)}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell \cdot -2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

      14. lower-/.f64 46.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(n + n\right) \cdot U\right)} \]

   10. Applied rewrites 46.9%

       \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \frac{-2}{Om}, \ell, t\right) \cdot \left(\left(\color{blue}{n} + n\right) \cdot U\right)} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.9%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ t_3 := \sqrt{\left(\left(U + U\right) \cdot \mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right)\right) \cdot n}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+30}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (fabs (* t (* U (+ n n))))))
        (t_3 (sqrt (* (* (+ U U) (fma (/ (* -2.0 l) Om) l t)) n))))
   (if (<= t_1 5e+30)
     t_3
     (if (<= t_1 1e+129) t_2 (if (<= t_1 INFINITY) t_3 t_2)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt(fabs((t * (U * (n + n)))));
	double t_3 = sqrt((((U + U) * fma(((-2.0 * l) / Om), l, t)) * n));
	double tmp;
	if (t_1 <= 5e+30) {
		tmp = t_3;
	} else if (t_1 <= 1e+129) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))))
	t_3 = sqrt(Float64(Float64(Float64(U + U) * fma(Float64(Float64(-2.0 * l) / Om), l, t)) * n))
	tmp = 0.0
	if (t_1 <= 5e+30)
		tmp = t_3;
	elseif (t_1 <= 1e+129)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e+30], t$95$3, If[LessEqual[t$95$1, 1e+129], t$95$2, If[LessEqual[t$95$1, Infinity], t$95$3, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999998e30 or 1e129 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 47.1%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot \mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right)\right) \cdot \color{blue}{n}} \]

   if 4.9999999999999998e30 < (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*))))) < 1e129 or +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.9%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      INFINITY)
   (sqrt (* (+ U U) (* (fma (/ (* -2.0 l) Om) l t) n)))
   (sqrt (fabs (* 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_))))) <= ((double) INFINITY)) {
		tmp = sqrt(((U + U) * (fma(((-2.0 * l) / Om), l, t) * n)));
	} else {
		tmp = sqrt(fabs((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_))))) <= Inf)
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(Float64(Float64(-2.0 * l) / Om), l, t) * n)));
	else
		tmp = sqrt(abs(Float64(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], Infinity], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $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*))))) < +inf.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\frac{\ell \cdot \ell}{Om} + \frac{\ell \cdot \ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\frac{\ell \cdot \ell}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}} + \frac{\ell \cdot \ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{Om} + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. sqr-neg-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\mathsf{neg}\left(\ell\right)\right)\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right)} \cdot \left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\ell\right)\right) \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\color{blue}{\left(-\ell\right)} \cdot \left(\frac{1}{Om} + \frac{1}{Om}\right)\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. div-add-rev N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{1 + 1}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{\color{blue}{2}}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower-/.f64 53.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \color{blue}{\frac{2}{Om}}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{\left(-\ell\right) \cdot \left(\left(-\ell\right) \cdot \frac{2}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)\right)} \]

      7. lower-pow.f64 44.0

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

   6. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

   8. Applied rewrites 48.2%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-2 \cdot \ell}{Om}, \ell, t\right) \cdot n\right)}} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.9%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 40.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (+ U U) (* t n)))
   (sqrt (fabs (* 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 + U) * (t * n)));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt(((u + u) * (t * n)))
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt(((U + U) * (t * n)))
	else:
		tmp = math.sqrt(math.fabs((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(Float64(U + U) * Float64(t * n)));
	else
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 0.0)
		tmp = sqrt(((U + U) * (t * n)));
	else
		tmp = sqrt(abs((t * (U * (n + n)))));
	end
	tmp_2 = 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[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $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 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-+.f64 35.3

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 35.3

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 35.3%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.9%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \]

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*)))))
      5e-160)
   (sqrt (* (+ U U) (* t n)))
   (sqrt (* (* U (+ n n)) t))))

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_))))) <= 5e-160) {
		tmp = sqrt(((U + U) * (t * n)));
	} else {
		tmp = sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 5d-160) then
        tmp = sqrt(((u + u) * (t * n)))
    else
        tmp = sqrt(((u * (n + n)) * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 5e-160) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 5e-160:
		tmp = math.sqrt(((U + U) * (t * n)))
	else:
		tmp = math.sqrt(((U * (n + n)) * t))
	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_))))) <= 5e-160)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 5e-160)
		tmp = sqrt(((U + U) * (t * n)));
	else
		tmp = sqrt(((U * (n + n)) * t));
	end
	tmp_2 = 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], 5e-160], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.99999999999999994e-160

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-+.f64 35.3

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 35.3

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 35.3%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 4.99999999999999994e-160 < (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 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      7. associate-*l* 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. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      10. lower-*.f64 35.1

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lower-*.f64 35.1

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      15. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      16. lower-+.f64 35.1

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 35.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 37.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \]

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 U) t) n))
   (sqrt (* (* U (+ n n)) t))))

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 + U) * t) * n));
	} else {
		tmp = sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt(((u * (n + n)) * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt((((U + U) * t) * n))
	else:
		tmp = math.sqrt(((U * (n + n)) * t))
	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(U + U) * t) * n));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 0.0)
		tmp = sqrt((((U + U) * t) * n));
	else
		tmp = sqrt(((U * (n + n)) * t));
	end
	tmp_2 = 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[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

      10. lower-+.f64 34.6

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

   6. Applied rewrites 34.6%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

   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 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.3

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      7. associate-*l* 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. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      10. lower-*.f64 35.1

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lower-*.f64 35.1

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      15. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      16. lower-+.f64 35.1

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 35.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.1% accurate, 4.7× speedup

Math

\[\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

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(Float64(U * 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[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

2. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lower-*.f64 35.3

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 35.3%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   7. associate-*l* 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. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   10. lower-*.f64 35.1

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   13. lower-*.f64 35.1

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   15. count-2-rev N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   16. lower-+.f64 35.1

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

6. Applied rewrites 35.1%

   \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025176 
(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*))))))