Toniolo and Linder, Equation (13)

Percentage Accurate: 50.7% → 64.2%

Time: 17.3s

Alternatives: 28

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot t\_3\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_4 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(t\_1 - \left(t\_3 \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{t\_4}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om}, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l Om) l) -2.0 t))
        (t_2 (* (* 2.0 n) U))
        (t_3 (pow (/ l Om) 2.0))
        (t_4 (* t_2 (- (- t (* 2.0 (/ (* l l) Om))) (* (* n t_3) (- U U*))))))
   (if (<= t_4 4e-320)
     (* (sqrt (* (- t_1 (* (* t_3 n) (- U U*))) (* 2.0 n))) (sqrt U))
     (if (<= t_4 4e+268)
       (sqrt t_4)
       (if (<= t_4 INFINITY)
         (sqrt (* t_2 (fma (* (* (- n) (- U U*)) (/ l Om)) (/ l Om) t_1)))
         (sqrt
          (*
           -2.0
           (fma
            (* (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n) l)
            (* U l)
            (* (- n) (* U t))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(((l / Om) * l), -2.0, t);
	double t_2 = (2.0 * n) * U;
	double t_3 = pow((l / Om), 2.0);
	double t_4 = t_2 * ((t - (2.0 * ((l * l) / Om))) - ((n * t_3) * (U - U_42_)));
	double tmp;
	if (t_4 <= 4e-320) {
		tmp = sqrt(((t_1 - ((t_3 * n) * (U - U_42_))) * (2.0 * n))) * sqrt(U);
	} else if (t_4 <= 4e+268) {
		tmp = sqrt(t_4);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(((-n * (U - U_42_)) * (l / Om)), (l / Om), t_1)));
	} else {
		tmp = sqrt((-2.0 * fma((((fma((n / Om), (U - U_42_), 2.0) / Om) * n) * l), (U * l), (-n * (U * t)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(Float64(l / Om) * l), -2.0, t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(l / Om) ^ 2.0
	t_4 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * t_3) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_4 <= 4e-320)
		tmp = Float64(sqrt(Float64(Float64(t_1 - Float64(Float64(t_3 * n) * Float64(U - U_42_))) * Float64(2.0 * n))) * sqrt(U));
	elseif (t_4 <= 4e+268)
		tmp = sqrt(t_4);
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(Float64(-n) * Float64(U - U_42_)) * Float64(l / Om)), Float64(l / Om), t_1)));
	else
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n) * l), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * t$95$3), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 4e-320], N[(N[Sqrt[N[(N[(t$95$1 - N[(N[(t$95$3 * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+268], N[Sqrt[t$95$4], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $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*)))) < 3.99996e-320

   1. Initial program 12.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 14.5

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 14.5%

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

   6. Applied rewrites 44.4%

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

   if 3.99996e-320 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999999e268

   1. Initial program 99.1%

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

   if 3.9999999999999999e268 < (*.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 34.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 45.5

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 44.9%

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

   6. Applied rewrites 46.0%

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

   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 0.0%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 32.5%

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

   7. Applied rewrites 55.4%

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

Alternative 2: 64.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t\_1, \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{t\_3}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(t\_1, \frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* (- n) (- U U*)) (/ l Om)))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 4e-290)
     (sqrt (* (* (fma t_1 (/ l Om) (fma (* -2.0 (/ l Om)) l t)) (* 2.0 n)) U))
     (if (<= t_3 4e+268)
       (sqrt t_3)
       (if (<= t_3 INFINITY)
         (sqrt (* t_2 (fma t_1 (/ l Om) (fma (* (/ l Om) l) -2.0 t))))
         (sqrt
          (*
           -2.0
           (fma
            (* (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n) l)
            (* U l)
            (* (- n) (* U t))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (-n * (U - U_42_)) * (l / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 4e-290) {
		tmp = sqrt(((fma(t_1, (l / Om), fma((-2.0 * (l / Om)), l, t)) * (2.0 * n)) * U));
	} else if (t_3 <= 4e+268) {
		tmp = sqrt(t_3);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(t_1, (l / Om), fma(((l / Om) * l), -2.0, t))));
	} else {
		tmp = sqrt((-2.0 * fma((((fma((n / Om), (U - U_42_), 2.0) / Om) * n) * l), (U * l), (-n * (U * t)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(-n) * Float64(U - U_42_)) * Float64(l / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 4e-290)
		tmp = sqrt(Float64(Float64(fma(t_1, Float64(l / Om), fma(Float64(-2.0 * Float64(l / Om)), l, t)) * Float64(2.0 * n)) * U));
	elseif (t_3 <= 4e+268)
		tmp = sqrt(t_3);
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(t_1, Float64(l / Om), fma(Float64(Float64(l / Om) * l), -2.0, t))));
	else
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n) * l), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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$3, 4e-290], N[Sqrt[N[(N[(N[(t$95$1 * N[(l / Om), $MachinePrecision] + N[(N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 4e+268], N[Sqrt[t$95$3], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(t$95$1 * N[(l / Om), $MachinePrecision] + N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $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*)))) < 4.0000000000000003e-290

   1. Initial program 14.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 17.0

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 17.0%

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

   6. Applied rewrites 42.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 42.2%

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

   if 4.0000000000000003e-290 < (*.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*)))) < 3.9999999999999999e268

   1. Initial program 99.7%

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

   if 3.9999999999999999e268 < (*.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 34.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 45.5

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 44.9%

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

   6. Applied rewrites 46.0%

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

   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 0.0%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 32.5%

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

   7. Applied rewrites 55.4%

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

Alternative 3: 64.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_4 := t\_2 \cdot \left(t\_3 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_4 \leq 4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t\_1, \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t\_3 - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(t\_1, \frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* (- n) (- U U*)) (/ l Om)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (- t (* 2.0 (/ (* l l) Om))))
        (t_4 (* t_2 (- t_3 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_4 4e-290)
     (sqrt (* (* (fma t_1 (/ l Om) (fma (* -2.0 (/ l Om)) l t)) (* 2.0 n)) U))
     (if (<= t_4 4e+268)
       (sqrt (* t_2 (- t_3 (* (* (- U U*) (/ l Om)) (* (/ l Om) n)))))
       (if (<= t_4 INFINITY)
         (sqrt (* t_2 (fma t_1 (/ l Om) (fma (* (/ l Om) l) -2.0 t))))
         (sqrt
          (*
           -2.0
           (fma
            (* (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n) l)
            (* U l)
            (* (- n) (* U t))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (-n * (U - U_42_)) * (l / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = t - (2.0 * ((l * l) / Om));
	double t_4 = t_2 * (t_3 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_4 <= 4e-290) {
		tmp = sqrt(((fma(t_1, (l / Om), fma((-2.0 * (l / Om)), l, t)) * (2.0 * n)) * U));
	} else if (t_4 <= 4e+268) {
		tmp = sqrt((t_2 * (t_3 - (((U - U_42_) * (l / Om)) * ((l / Om) * n)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(t_1, (l / Om), fma(((l / Om) * l), -2.0, t))));
	} else {
		tmp = sqrt((-2.0 * fma((((fma((n / Om), (U - U_42_), 2.0) / Om) * n) * l), (U * l), (-n * (U * t)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(-n) * Float64(U - U_42_)) * Float64(l / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_4 = Float64(t_2 * Float64(t_3 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_4 <= 4e-290)
		tmp = sqrt(Float64(Float64(fma(t_1, Float64(l / Om), fma(Float64(-2.0 * Float64(l / Om)), l, t)) * Float64(2.0 * n)) * U));
	elseif (t_4 <= 4e+268)
		tmp = sqrt(Float64(t_2 * Float64(t_3 - Float64(Float64(Float64(U - U_42_) * Float64(l / Om)) * Float64(Float64(l / Om) * n)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(t_1, Float64(l / Om), fma(Float64(Float64(l / Om) * l), -2.0, t))));
	else
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n) * l), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$3 - 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$4, 4e-290], N[Sqrt[N[(N[(N[(t$95$1 * N[(l / Om), $MachinePrecision] + N[(N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 4e+268], N[Sqrt[N[(t$95$2 * N[(t$95$3 - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$2 * N[(t$95$1 * N[(l / Om), $MachinePrecision] + N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $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*)))) < 4.0000000000000003e-290

   1. Initial program 14.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 17.0

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 17.0%

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

   6. Applied rewrites 42.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 42.2%

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

   if 4.0000000000000003e-290 < (*.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*)))) < 3.9999999999999999e268

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if 3.9999999999999999e268 < (*.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 34.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 45.5

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 44.9%

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

   6. Applied rewrites 46.0%

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

   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 0.0%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 32.5%

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

   7. Applied rewrites 55.4%

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

Alternative 4: 48.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 10^{+297}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(\frac{-\ell}{Om} \cdot \sqrt{U \cdot \left(U* - U\right)}\right) \cdot \left(n \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot n\right) \cdot \left(\frac{\sqrt{\left(U* - U\right) \cdot U}}{Om} \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 4e-320)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 1e+297)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         (* (* (/ (- l) Om) (sqrt (* U (- U* U)))) (* n (sqrt 2.0)))
         (* (* (sqrt 2.0) n) (* (/ (sqrt (* (- U* U) U)) Om) l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 4e-320) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= 1e+297) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = ((-l / Om) * sqrt((U * (U_42_ - U)))) * (n * sqrt(2.0));
	} else {
		tmp = (sqrt(2.0) * n) * ((sqrt(((U_42_ - U) * U)) / Om) * l);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 4e-320)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= 1e+297)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(Float64(-l) / Om) * sqrt(Float64(U * Float64(U_42_ - U)))) * Float64(n * sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) * n) * Float64(Float64(sqrt(Float64(Float64(U_42_ - U) * U)) / Om) * l));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 4e-320], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+297], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[((-l) / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(U$42$ - U), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision] * l), $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*)))) < 3.99996e-320

   1. Initial program 12.4%

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

   4. Applied rewrites 31.4%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   if 3.99996e-320 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e297

   1. Initial program 99.1%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 1e297 < (*.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 31.7%

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

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

   8. Taylor expanded in l around inf

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

   10. Applied rewrites 12.5%

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

   11. Taylor expanded in Om around -inf

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

   13. Applied rewrites 21.5%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 13.2

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

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in l around inf

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

   10. Applied rewrites 6.8%

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

   12. Applied rewrites 30.0%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+297}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\left(\frac{-\ell}{Om} \cdot \sqrt{U \cdot \left(U* - U\right)}\right) \cdot \left(n \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot n\right) \cdot \left(\frac{\sqrt{\left(U* - U\right) \cdot U}}{Om} \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 5e-145)
		tmp = sqrt(Float64(fma(Float64(Float64(l * l) * U), Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(n / Om)), Float64(Float64(Float64(-n) * t) * U)) * -2.0));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(Float64(Float64(Float64(-n) * Float64(U - U_42_)) * Float64(l / Om)), Float64(l / Om), fma(Float64(Float64(l / Om) * l), -2.0, t))));
	else
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n) * l), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	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[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 5e-145], N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] + N[(N[((-n) * t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $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*))))) < 4.9999999999999998e-145

   1. Initial program 17.6%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 42.0%

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

   9. Applied rewrites 43.9%

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

   if 4.9999999999999998e-145 < (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 64.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 71.0

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 66.6%

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

   6. Applied rewrites 67.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 30.0%

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

   7. Applied rewrites 51.2%

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

Alternative 6: 62.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t\_1, \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(t\_1, \frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* (- n) (- U U*)) (/ l Om)))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 4e-290)
     (sqrt (* (* (fma t_1 (/ l Om) (fma (* -2.0 (/ l Om)) l t)) (* 2.0 n)) U))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (fma t_1 (/ l Om) (fma (* (/ l Om) l) -2.0 t))))
       (sqrt
        (*
         -2.0
         (fma
          (* (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n) l)
          (* U l)
          (* (- n) (* U t)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (-n * (U - U_42_)) * (l / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 4e-290) {
		tmp = sqrt(((fma(t_1, (l / Om), fma((-2.0 * (l / Om)), l, t)) * (2.0 * n)) * U));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(t_1, (l / Om), fma(((l / Om) * l), -2.0, t))));
	} else {
		tmp = sqrt((-2.0 * fma((((fma((n / Om), (U - U_42_), 2.0) / Om) * n) * l), (U * l), (-n * (U * t)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(-n) * Float64(U - U_42_)) * Float64(l / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 4e-290)
		tmp = sqrt(Float64(Float64(fma(t_1, Float64(l / Om), fma(Float64(-2.0 * Float64(l / Om)), l, t)) * Float64(2.0 * n)) * U));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(t_1, Float64(l / Om), fma(Float64(Float64(l / Om) * l), -2.0, t))));
	else
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n) * l), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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$3, 4e-290], N[Sqrt[N[(N[(N[(t$95$1 * N[(l / Om), $MachinePrecision] + N[(N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l + t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(t$95$1 * N[(l / Om), $MachinePrecision] + N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $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*)))) < 4.0000000000000003e-290

   1. Initial program 14.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 17.0

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 17.0%

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

   6. Applied rewrites 42.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 42.2%

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

   if 4.0000000000000003e-290 < (*.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 65.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 71.2

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 66.8%

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

   6. Applied rewrites 67.4%

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

   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 0.0%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 32.5%

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

   7. Applied rewrites 55.4%

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

Alternative 7: 48.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \mathsf{fma}\left(-2, t\_1, t\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\left(\left(t\_2 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(2, n, t\right)\right)} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (fma -2.0 t_1 t))
        (t_3 (* (* 2.0 n) U))
        (t_4
         (sqrt
          (*
           t_3
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_4 2e-145)
     (sqrt (* (* (* t_2 n) U) 2.0))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 t_2))
       (* (sqrt (* U (* n (fma 2.0 n t)))) (sqrt 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = fma(-2.0, t_1, t);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_4 <= 2e-145) {
		tmp = sqrt((((t_2 * n) * U) * 2.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * t_2));
	} else {
		tmp = sqrt((U * (n * fma(2.0, n, t)))) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = fma(-2.0, t_1, t)
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_4 <= 2e-145)
		tmp = sqrt(Float64(Float64(Float64(t_2 * n) * U) * 2.0));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * t_2));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * fma(2.0, n, t)))) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * t$95$1 + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $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$4, 2e-145], N[Sqrt[N[(N[(N[(t$95$2 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(2.0 * n + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $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*))))) < 1.99999999999999983e-145

   1. Initial program 15.9%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

   if 1.99999999999999983e-145 < (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 65.1%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 20.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sqrt.f64 18.2

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

   7. Applied rewrites 18.2%

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

Alternative 8: 48.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot n\right) \cdot \left(\frac{\sqrt{\left(U* - U\right) \cdot U}}{Om} \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 4e-320)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (* (* (sqrt 2.0) n) (* (/ (sqrt (* (- U* U) U)) Om) l))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 4e-320) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = (sqrt(2.0) * n) * ((sqrt(((U_42_ - U) * U)) / Om) * l);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 4e-320)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(Float64(sqrt(2.0) * n) * Float64(Float64(sqrt(Float64(Float64(U_42_ - U) * U)) / Om) * l));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 4e-320], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(U$42$ - U), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision] * l), $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*)))) < 3.99996e-320

   1. Initial program 12.4%

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

   4. Applied rewrites 31.4%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   if 3.99996e-320 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 65.2%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 13.2

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

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in l around inf

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

   10. Applied rewrites 6.8%

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

   12. Applied rewrites 30.0%

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

Alternative 9: 48.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{Om} \cdot \sqrt{U \cdot U*}\right) \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 4e-320)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (* (* (/ l Om) (sqrt (* U U*))) (* n (sqrt 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 4e-320) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = ((l / Om) * sqrt((U * U_42_))) * (n * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 4e-320)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(Float64(Float64(l / Om) * sqrt(Float64(U * U_42_))) * Float64(n * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 4e-320], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(l / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.99996e-320

   1. Initial program 12.4%

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

   4. Applied rewrites 31.4%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   if 3.99996e-320 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 65.2%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 13.2

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

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in U* around inf

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

   10. Applied rewrites 24.7%

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

Alternative 10: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 4e-320)
     (* (sqrt (* (* 2.0 n) t)) (sqrt U))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 4e-320) {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 4e-320)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 4e-320], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / 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*)))) < 3.99996e-320

   1. Initial program 12.4%

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

   4. Applied rewrites 31.4%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   if 3.99996e-320 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 65.2%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 22.4

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

   5. Applied rewrites 22.4%

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

Alternative 11: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, t\_1, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \left(\ell \cdot U\right) \cdot \left(\ell \cdot \frac{n}{Om}\right), 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om)))
   (if (<=
        (sqrt
         (*
          (* (* 2.0 n) U)
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        2e-126)
     (sqrt (* (* (* (fma -2.0 t_1 t) n) U) 2.0))
     (sqrt (fma -4.0 (* (* l U) (* l (/ n Om))) (* 2.0 (* (* U n) t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 2e-126) {
		tmp = sqrt((((fma(-2.0, t_1, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(fma(-4.0, ((l * U) * (l * (n / Om))), (2.0 * ((U * n) * t))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 2e-126)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, t_1, t) * n) * U) * 2.0));
	else
		tmp = sqrt(fma(-4.0, Float64(Float64(l * U) * Float64(l * Float64(n / Om))), Float64(2.0 * Float64(Float64(U * n) * t))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-126], N[Sqrt[N[(N[(N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(l * U), $MachinePrecision] * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $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*))))) < 1.9999999999999999e-126

   1. Initial program 23.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 46.4

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

   5. Applied rewrites 46.4%

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

   if 1.9999999999999999e-126 < (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 52.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 60.2

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.8

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

   7. Applied rewrites 41.8%

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

   9. Applied rewrites 48.7%

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

Alternative 12: 39.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      2e-160)
   (sqrt (* (* (* U t) n) 2.0))
   (sqrt (* t (* (* U n) 2.0)))))

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_))))) <= 2e-160) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = sqrt((t * ((U * n) * 2.0)));
	}
	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))))) <= 2d-160) then
        tmp = sqrt((((u * t) * n) * 2.0d0))
    else
        tmp = sqrt((t * ((u * n) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 2e-160) {
		tmp = Math.sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = Math.sqrt((t * ((U * n) * 2.0)));
	}
	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_))))) <= 2e-160:
		tmp = math.sqrt((((U * t) * n) * 2.0))
	else:
		tmp = math.sqrt((t * ((U * n) * 2.0)))
	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_))))) <= 2e-160)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	else
		tmp = sqrt(Float64(t * Float64(Float64(U * n) * 2.0)));
	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_))))) <= 2e-160)
		tmp = sqrt((((U * t) * n) * 2.0));
	else
		tmp = sqrt((t * ((U * n) * 2.0)));
	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], 2e-160], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * N[(N[(U * n), $MachinePrecision] * 2.0), $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*))))) < 2e-160

   1. Initial program 13.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.1

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 36.3%

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

   if 2e-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 53.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 36.6%

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

Alternative 13: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.8 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\left(\ell \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right) \cdot n, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot U} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 8.8e-119)
   (sqrt
    (*
     -2.0
     (fma
      (* (* l (/ (fma (- U U*) (/ n Om) 2.0) Om)) n)
      (* U l)
      (* (- n) (* U t)))))
   (*
    (sqrt
     (*
      (- (fma (* (/ l Om) l) -2.0 t) (* (* (pow (/ l Om) 2.0) n) (- U U*)))
      U))
    (sqrt (* 2.0 n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 8.8e-119) {
		tmp = sqrt((-2.0 * fma(((l * (fma((U - U_42_), (n / Om), 2.0) / Om)) * n), (U * l), (-n * (U * t)))));
	} else {
		tmp = sqrt(((fma(((l / Om) * l), -2.0, t) - ((pow((l / Om), 2.0) * n) * (U - U_42_))) * U)) * sqrt((2.0 * n));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 8.8e-119)
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(l * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om)) * n), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) - Float64(Float64((Float64(l / Om) ^ 2.0) * n) * Float64(U - U_42_))) * U)) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 8.8e-119], N[Sqrt[N[(-2.0 * N[(N[(N[(l * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] - N[(N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.8000000000000002e-119

   1. Initial program 42.5%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 53.2%

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

   9. Applied rewrites 55.6%

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

   if 8.8000000000000002e-119 < n

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 58.8

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 69.4%

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

Alternative 14: 51.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)\\ \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot U, t\_1 \cdot \frac{n}{Om}, \left(\left(-n\right) \cdot t\right) \cdot U\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\left(\ell \cdot \frac{t\_1}{Om}\right) \cdot n, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (- U U*) (/ n Om) 2.0)))
   (if (<= l 1.1e-140)
     (sqrt (* (* (fma -2.0 (/ (* l l) Om) t) (* 2.0 n)) U))
     (if (<= l 1.75e+29)
       (sqrt (* (fma (* (* l l) U) (* t_1 (/ n Om)) (* (* (- n) t) U)) -2.0))
       (sqrt
        (* -2.0 (fma (* (* l (/ t_1 Om)) n) (* U l) (* (- n) (* U t)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((U - U_42_), (n / Om), 2.0);
	double tmp;
	if (l <= 1.1e-140) {
		tmp = sqrt(((fma(-2.0, ((l * l) / Om), t) * (2.0 * n)) * U));
	} else if (l <= 1.75e+29) {
		tmp = sqrt((fma(((l * l) * U), (t_1 * (n / Om)), ((-n * t) * U)) * -2.0));
	} else {
		tmp = sqrt((-2.0 * fma(((l * (t_1 / Om)) * n), (U * l), (-n * (U * t)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(U - U_42_), Float64(n / Om), 2.0)
	tmp = 0.0
	if (l <= 1.1e-140)
		tmp = sqrt(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * Float64(2.0 * n)) * U));
	elseif (l <= 1.75e+29)
		tmp = sqrt(Float64(fma(Float64(Float64(l * l) * U), Float64(t_1 * Float64(n / Om)), Float64(Float64(Float64(-n) * t) * U)) * -2.0));
	else
		tmp = sqrt(Float64(-2.0 * fma(Float64(Float64(l * Float64(t_1 / Om)) * n), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[l, 1.1e-140], N[Sqrt[N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.75e+29], N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] * U), $MachinePrecision] * N[(t$95$1 * N[(n / Om), $MachinePrecision]), $MachinePrecision] + N[(N[((-n) * t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(l * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.1e-140

   1. Initial program 49.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 54.0

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 54.4%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 45.5

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

   9. Applied rewrites 45.5%

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

   if 1.1e-140 < l < 1.74999999999999989e29

   1. Initial program 55.7%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 52.7%

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

   9. Applied rewrites 58.6%

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

   if 1.74999999999999989e29 < l

   1. Initial program 33.6%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 42.0%

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 64.1%

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

Alternative 15: 49.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 - \left(\frac{U - U*}{Om} \cdot \frac{\ell}{Om}\right) \cdot \ell\right)} \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.2e+167)
   (sqrt (/ (/ (* (* (* (* n l) U) (* n l)) (* (- U U*) -2.0)) Om) Om))
   (if (<= n 2.5e+122)
     (sqrt (* -2.0 (fma (* 2.0 (/ (* l n) Om)) (* U l) (* (- n) (* U t)))))
     (*
      (sqrt (* U (- 2.0 (* (* (/ (- U U*) Om) (/ l Om)) l))))
      (* n (sqrt 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.2e+167) {
		tmp = sqrt(((((((n * l) * U) * (n * l)) * ((U - U_42_) * -2.0)) / Om) / Om));
	} else if (n <= 2.5e+122) {
		tmp = sqrt((-2.0 * fma((2.0 * ((l * n) / Om)), (U * l), (-n * (U * t)))));
	} else {
		tmp = sqrt((U * (2.0 - ((((U - U_42_) / Om) * (l / Om)) * l)))) * (n * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -2.2e+167)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(n * l) * U) * Float64(n * l)) * Float64(Float64(U - U_42_) * -2.0)) / Om) / Om));
	elseif (n <= 2.5e+122)
		tmp = sqrt(Float64(-2.0 * fma(Float64(2.0 * Float64(Float64(l * n) / Om)), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(2.0 - Float64(Float64(Float64(Float64(U - U_42_) / Om) * Float64(l / Om)) * l)))) * Float64(n * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -2.2e+167], N[Sqrt[N[(N[(N[(N[(N[(N[(n * l), $MachinePrecision] * U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.5e+122], N[Sqrt[N[(-2.0 * N[(N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(2.0 - N[(N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.20000000000000003e167

   1. Initial program 39.1%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 48.5%

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

   if -2.20000000000000003e167 < n < 2.49999999999999994e122

   1. Initial program 46.2%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 54.9%

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

   if 2.49999999999999994e122 < n

   1. Initial program 52.2%

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

   4. Applied rewrites 16.5%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 43.9

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 52.3%

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

Alternative 16: 51.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.8e-138)
   (sqrt (* (* (fma -2.0 (/ (* l l) Om) t) (* 2.0 n)) U))
   (sqrt
    (*
     -2.0
     (fma
      (* (* l (/ (fma (- U U*) (/ n Om) 2.0) Om)) n)
      (* U l)
      (* (- n) (* U t)))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < 4.7999999999999998e-138

   1. Initial program 49.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 54.0

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 54.4%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 45.5

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

   9. Applied rewrites 45.5%

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

   if 4.7999999999999998e-138 < l

   1. Initial program 41.1%

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

   3. Taylor expanded in l around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 47.6%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 60.2%

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

Alternative 17: 48.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right), \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7e-93)
   (sqrt (fma (/ (* (* (* l l) n) U) Om) -4.0 (* (* (* n t) U) 2.0)))
   (sqrt
    (*
     -2.0
     (fma
      l
      (* l (* (* U n) (/ (fma (/ n Om) (- U U*) 2.0) Om)))
      (* (- n) (* U t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7e-93) {
		tmp = sqrt(fma(((((l * l) * n) * U) / Om), -4.0, (((n * t) * U) * 2.0)));
	} else {
		tmp = sqrt((-2.0 * fma(l, (l * ((U * n) * (fma((n / Om), (U - U_42_), 2.0) / Om))), (-n * (U * t)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7e-93)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l * l) * n) * U) / Om), -4.0, Float64(Float64(Float64(n * t) * U) * 2.0)));
	else
		tmp = sqrt(Float64(-2.0 * fma(l, Float64(l * Float64(Float64(U * n) * Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om))), Float64(Float64(-n) * Float64(U * t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7e-93], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(l * N[(l * N[(N[(U * n), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7e-93

   1. Initial program 49.0%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 44.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]

   if 7e-93 < l

   1. Initial program 41.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \color{blue}{-2} \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      3. distribute-lft-out-- N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied rewrites 48.7%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot n\right) - \left(n \cdot t\right) \cdot U\right)}} \]

   7. Applied rewrites 55.6%

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right)}, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}}\\ \mathbf{elif}\;n \leq 10^{+141}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 - \frac{\left(\ell \cdot \ell\right) \cdot \left(U - U*\right)}{Om \cdot Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.2e+167)
   (sqrt (/ (/ (* (* (* (* n l) U) (* n l)) (* (- U U*) -2.0)) Om) Om))
   (if (<= n 1e+141)
     (sqrt (* -2.0 (fma (* 2.0 (/ (* l n) Om)) (* U l) (* (- n) (* U t)))))
     (*
      (sqrt (* U (- 2.0 (/ (* (* l l) (- U U*)) (* Om Om)))))
      (* n (sqrt 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.2e+167) {
		tmp = sqrt(((((((n * l) * U) * (n * l)) * ((U - U_42_) * -2.0)) / Om) / Om));
	} else if (n <= 1e+141) {
		tmp = sqrt((-2.0 * fma((2.0 * ((l * n) / Om)), (U * l), (-n * (U * t)))));
	} else {
		tmp = sqrt((U * (2.0 - (((l * l) * (U - U_42_)) / (Om * Om))))) * (n * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -2.2e+167)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(n * l) * U) * Float64(n * l)) * Float64(Float64(U - U_42_) * -2.0)) / Om) / Om));
	elseif (n <= 1e+141)
		tmp = sqrt(Float64(-2.0 * fma(Float64(2.0 * Float64(Float64(l * n) / Om)), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(2.0 - Float64(Float64(Float64(l * l) * Float64(U - U_42_)) / Float64(Om * Om))))) * Float64(n * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -2.2e+167], N[Sqrt[N[(N[(N[(N[(N[(N[(n * l), $MachinePrecision] * U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1e+141], N[Sqrt[N[(-2.0 * N[(N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(2.0 - N[(N[(N[(l * l), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.20000000000000003e167

   1. Initial program 39.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot -2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot -2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}} \cdot -2} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      13. lower--.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \color{blue}{\left(U - U*\right)}\right)}{{Om}^{2}} \cdot -2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} \cdot -2} \]

      15. lower-*.f64 36.8

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} \cdot -2} \]

   5. Applied rewrites 36.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{Om \cdot Om} \cdot -2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \sqrt{\frac{\frac{\left(U \cdot {\left(\ell \cdot n\right)}^{2}\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{\color{blue}{Om}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 48.5%

      \[\leadsto \sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}} \]

   if -2.20000000000000003e167 < n < 1.00000000000000002e141

   1. Initial program 47.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \color{blue}{-2} \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      3. distribute-lft-out-- N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied rewrites 45.5%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot n\right) - \left(n \cdot t\right) \cdot U\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, \color{blue}{U \cdot \ell}, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.5%

       \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   if 1.00000000000000002e141 < n

   1. Initial program 47.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      6. times-frac N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om}} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \color{blue}{\frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{\color{blue}{U - U*}}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \color{blue}{\left(n \cdot \sqrt{2}\right)} \]

      14. lower-sqrt.f64 44.3

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 50.5%

      \[\leadsto \sqrt{U \cdot \left(2 - \frac{\left(\ell \cdot \ell\right) \cdot \left(U - U*\right)}{Om \cdot Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 48.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}}\\ \mathbf{elif}\;n \leq 10^{+141}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 + \frac{U* \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.2e+167)
   (sqrt (/ (/ (* (* (* (* n l) U) (* n l)) (* (- U U*) -2.0)) Om) Om))
   (if (<= n 1e+141)
     (sqrt (* -2.0 (fma (* 2.0 (/ (* l n) Om)) (* U l) (* (- n) (* U t)))))
     (* (sqrt (* U (+ 2.0 (/ (* U* (* l l)) (* Om Om))))) (* n (sqrt 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.2e+167) {
		tmp = sqrt(((((((n * l) * U) * (n * l)) * ((U - U_42_) * -2.0)) / Om) / Om));
	} else if (n <= 1e+141) {
		tmp = sqrt((-2.0 * fma((2.0 * ((l * n) / Om)), (U * l), (-n * (U * t)))));
	} else {
		tmp = sqrt((U * (2.0 + ((U_42_ * (l * l)) / (Om * Om))))) * (n * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -2.2e+167)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(n * l) * U) * Float64(n * l)) * Float64(Float64(U - U_42_) * -2.0)) / Om) / Om));
	elseif (n <= 1e+141)
		tmp = sqrt(Float64(-2.0 * fma(Float64(2.0 * Float64(Float64(l * n) / Om)), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(2.0 + Float64(Float64(U_42_ * Float64(l * l)) / Float64(Om * Om))))) * Float64(n * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -2.2e+167], N[Sqrt[N[(N[(N[(N[(N[(N[(n * l), $MachinePrecision] * U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1e+141], N[Sqrt[N[(-2.0 * N[(N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(2.0 + N[(N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.20000000000000003e167

   1. Initial program 39.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot -2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot -2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}} \cdot -2} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      13. lower--.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \color{blue}{\left(U - U*\right)}\right)}{{Om}^{2}} \cdot -2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} \cdot -2} \]

      15. lower-*.f64 36.8

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} \cdot -2} \]

   5. Applied rewrites 36.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{Om \cdot Om} \cdot -2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \sqrt{\frac{\frac{\left(U \cdot {\left(\ell \cdot n\right)}^{2}\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{\color{blue}{Om}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 48.5%

      \[\leadsto \sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}} \]

   if -2.20000000000000003e167 < n < 1.00000000000000002e141

   1. Initial program 47.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \color{blue}{-2} \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      3. distribute-lft-out-- N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied rewrites 45.5%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot n\right) - \left(n \cdot t\right) \cdot U\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, \color{blue}{U \cdot \ell}, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.5%

       \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   if 1.00000000000000002e141 < n

   1. Initial program 47.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      6. times-frac N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om}} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \color{blue}{\frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{\color{blue}{U - U*}}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \color{blue}{\left(n \cdot \sqrt{2}\right)} \]

      14. lower-sqrt.f64 44.3

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

   8. Taylor expanded in l around inf

      \[\leadsto \left(\sqrt{U \cdot \left(\frac{U*}{{Om}^{2}} - \frac{U}{{Om}^{2}}\right)} \cdot \ell\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(\sqrt{U \cdot \left(\frac{U*}{Om \cdot Om} - \frac{U}{Om \cdot Om}\right)} \cdot \ell\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]

   11. Taylor expanded in U around 0

       \[\leadsto \sqrt{U \cdot \left(2 - -1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 50.0%

       \[\leadsto \sqrt{U \cdot \left(2 - \left(-\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}\right)\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}}\\ \mathbf{elif}\;n \leq 10^{+141}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 + \frac{U* \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot \frac{U* - U}{Om \cdot Om}} \cdot \ell\right) \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -2.2e+167)
   (sqrt (/ (/ (* (* (* (* n l) U) (* n l)) (* (- U U*) -2.0)) Om) Om))
   (if (<= n 3.8e+140)
     (sqrt (* -2.0 (fma (* 2.0 (/ (* l n) Om)) (* U l) (* (- n) (* U t)))))
     (* (* (sqrt (* U (/ (- U* U) (* Om Om)))) l) (* n (sqrt 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -2.2e+167) {
		tmp = sqrt(((((((n * l) * U) * (n * l)) * ((U - U_42_) * -2.0)) / Om) / Om));
	} else if (n <= 3.8e+140) {
		tmp = sqrt((-2.0 * fma((2.0 * ((l * n) / Om)), (U * l), (-n * (U * t)))));
	} else {
		tmp = (sqrt((U * ((U_42_ - U) / (Om * Om)))) * l) * (n * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -2.2e+167)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(n * l) * U) * Float64(n * l)) * Float64(Float64(U - U_42_) * -2.0)) / Om) / Om));
	elseif (n <= 3.8e+140)
		tmp = sqrt(Float64(-2.0 * fma(Float64(2.0 * Float64(Float64(l * n) / Om)), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	else
		tmp = Float64(Float64(sqrt(Float64(U * Float64(Float64(U_42_ - U) / Float64(Om * Om)))) * l) * Float64(n * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -2.2e+167], N[Sqrt[N[(N[(N[(N[(N[(N[(n * l), $MachinePrecision] * U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.8e+140], N[Sqrt[N[(-2.0 * N[(N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.20000000000000003e167

   1. Initial program 39.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot -2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot -2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}} \cdot -2} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left({n}^{2} \cdot \left(U - U*\right)\right)}}{{Om}^{2}} \cdot -2} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot n\right)} \cdot \left(U - U*\right)\right)}{{Om}^{2}} \cdot -2} \]

      13. lower--.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \color{blue}{\left(U - U*\right)}\right)}{{Om}^{2}} \cdot -2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} \cdot -2} \]

      15. lower-*.f64 36.8

          \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} \cdot -2} \]

   5. Applied rewrites 36.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)}{Om \cdot Om} \cdot -2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \sqrt{\frac{\frac{\left(U \cdot {\left(\ell \cdot n\right)}^{2}\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{\color{blue}{Om}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 48.5%

      \[\leadsto \sqrt{\frac{\frac{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(\left(U - U*\right) \cdot -2\right)}{Om}}{Om}} \]

   if -2.20000000000000003e167 < n < 3.8000000000000001e140

   1. Initial program 47.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \color{blue}{-2} \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      3. distribute-lft-out-- N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied rewrites 45.5%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot n\right) - \left(n \cdot t\right) \cdot U\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, \color{blue}{U \cdot \ell}, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.5%

       \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   if 3.8000000000000001e140 < n

   1. Initial program 47.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      6. times-frac N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om}} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \color{blue}{\frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{\color{blue}{U - U*}}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \color{blue}{\left(n \cdot \sqrt{2}\right)} \]

      14. lower-sqrt.f64 44.3

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

   8. Taylor expanded in l around inf

      \[\leadsto \left(\sqrt{U \cdot \left(\frac{U*}{{Om}^{2}} - \frac{U}{{Om}^{2}}\right)} \cdot \ell\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(\sqrt{U \cdot \left(\frac{U*}{Om \cdot Om} - \frac{U}{Om \cdot Om}\right)} \cdot \ell\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 32.1%

       \[\leadsto \left(\sqrt{U \cdot \frac{U* - U}{Om \cdot Om}} \cdot \ell\right) \cdot \left(n \cdot \sqrt{2}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 47.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.8 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, U \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U \cdot \frac{U* - U}{Om \cdot Om}} \cdot \ell\right) \cdot \left(n \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 3.8e+140)
   (sqrt (* -2.0 (fma (* 2.0 (/ (* l n) Om)) (* U l) (* (- n) (* U t)))))
   (* (* (sqrt (* U (/ (- U* U) (* Om Om)))) l) (* n (sqrt 2.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 3.8e+140) {
		tmp = sqrt((-2.0 * fma((2.0 * ((l * n) / Om)), (U * l), (-n * (U * t)))));
	} else {
		tmp = (sqrt((U * ((U_42_ - U) / (Om * Om)))) * l) * (n * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 3.8e+140)
		tmp = sqrt(Float64(-2.0 * fma(Float64(2.0 * Float64(Float64(l * n) / Om)), Float64(U * l), Float64(Float64(-n) * Float64(U * t)))));
	else
		tmp = Float64(Float64(sqrt(Float64(U * Float64(Float64(U_42_ - U) / Float64(Om * Om)))) * l) * Float64(n * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 3.8e+140], N[Sqrt[N[(-2.0 * N[(N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision] + N[((-n) * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.8000000000000001e140

   1. Initial program 46.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) - \color{blue}{-2} \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      3. distribute-lft-out-- N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) - U \cdot \left(n \cdot t\right)\right)}} \]

   5. Applied rewrites 44.0%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right) \cdot n\right) - \left(n \cdot t\right) \cdot U\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.6%

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right) \cdot \ell, \color{blue}{U \cdot \ell}, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.6%

       \[\leadsto \sqrt{-2 \cdot \mathsf{fma}\left(2 \cdot \frac{\ell \cdot n}{Om}, \color{blue}{U} \cdot \ell, \left(-n\right) \cdot \left(U \cdot t\right)\right)} \]

   if 3.8000000000000001e140 < n

   1. Initial program 47.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      6. times-frac N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om}} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \color{blue}{\frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{\color{blue}{U - U*}}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \color{blue}{\left(n \cdot \sqrt{2}\right)} \]

      14. lower-sqrt.f64 44.3

          \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

   8. Taylor expanded in l around inf

      \[\leadsto \left(\sqrt{U \cdot \left(\frac{U*}{{Om}^{2}} - \frac{U}{{Om}^{2}}\right)} \cdot \ell\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(\sqrt{U \cdot \left(\frac{U*}{Om \cdot Om} - \frac{U}{Om \cdot Om}\right)} \cdot \ell\right) \cdot \left(\color{blue}{n} \cdot \sqrt{2}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 32.1%

       \[\leadsto \left(\sqrt{U \cdot \frac{U* - U}{Om \cdot Om}} \cdot \ell\right) \cdot \left(n \cdot \sqrt{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 44.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{+196}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 1e+196)
   (sqrt (* (* (fma -2.0 (/ (* l l) Om) t) (* 2.0 n)) U))
   (sqrt (* (* (* U t) n) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 1e+196) {
		tmp = sqrt(((fma(-2.0, ((l * l) / Om), t) * (2.0 * n)) * U));
	} else {
		tmp = sqrt((((U * t) * n) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 1e+196)
		tmp = sqrt(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * Float64(2.0 * n)) * U));
	else
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 1e+196], N[Sqrt[N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 9.9999999999999995e195

   1. Initial program 48.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om} + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. associate--l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \ell} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{-2} \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. lower--.f64 53.9

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, \color{blue}{t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   4. Applied rewrites 51.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   6. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \left(2 \cdot n\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot \left(2 \cdot n\right)\right) \cdot U} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot \left(2 \cdot n\right)\right) \cdot U} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U} \]

      5. lower-*.f64 45.1

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U} \]

   9. Applied rewrites 45.1%

      \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \cdot \left(2 \cdot n\right)\right) \cdot U} \]

   if 9.9999999999999995e195 < Om

   1. Initial program 28.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 30.0

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 30.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.6%

      \[\leadsto \sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 44.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{+196}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 1e+196)
   (sqrt (* (* (* (fma -2.0 (/ (* l l) Om) t) n) U) 2.0))
   (sqrt (* (* (* U t) n) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 1e+196) {
		tmp = sqrt((((fma(-2.0, ((l * l) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((U * t) * n) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 1e+196)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l * l) / Om), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 1e+196], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 9.9999999999999995e195

   1. Initial program 48.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 45.1

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   5. Applied rewrites 45.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 9.9999999999999995e195 < Om

   1. Initial program 28.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 30.0

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 30.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.6%

      \[\leadsto \sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 39.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -5e-311)
   (sqrt (* (* (* n t) U) 2.0))
   (* (sqrt (* (* 2.0 n) t)) (sqrt U))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -5e-311) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	}
	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 (u <= (-5d-311)) then
        tmp = sqrt((((n * t) * u) * 2.0d0))
    else
        tmp = sqrt(((2.0d0 * n) * t)) * sqrt(u)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -5e-311) {
		tmp = Math.sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * n) * t)) * Math.sqrt(U);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -5e-311:
		tmp = math.sqrt((((n * t) * U) * 2.0))
	else:
		tmp = math.sqrt(((2.0 * n) * t)) * math.sqrt(U)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -5e-311)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -5e-311)
		tmp = sqrt((((n * t) * U) * 2.0));
	else
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -5e-311], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -5.00000000000023e-311

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 43.2

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 43.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   if -5.00000000000023e-311 < U

   1. Initial program 45.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot t\right)}} \cdot \sqrt{U} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot t}} \cdot \sqrt{U} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot t}} \cdot \sqrt{U} \]

      3. lower-*.f64 39.1

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot t} \cdot \sqrt{U} \]

   7. Applied rewrites 39.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot t}} \cdot \sqrt{U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 39.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{U \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5e-310)
   (sqrt (* (* (* n t) U) 2.0))
   (* (sqrt (* 2.0 t)) (sqrt (* U n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e-310) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = sqrt((2.0 * t)) * sqrt((U * n));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = sqrt((((n * t) * u) * 2.0d0))
    else
        tmp = sqrt((2.0d0 * t)) * sqrt((u * n))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e-310) {
		tmp = Math.sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = Math.sqrt((2.0 * t)) * Math.sqrt((U * n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5e-310:
		tmp = math.sqrt((((n * t) * U) * 2.0))
	else:
		tmp = math.sqrt((2.0 * t)) * math.sqrt((U * n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(2.0 * t)) * sqrt(Float64(U * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = sqrt((((n * t) * U) * 2.0));
	else
		tmp = sqrt((2.0 * t)) * sqrt((U * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5e-310], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 44.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 35.3

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   if -4.999999999999985e-310 < t

   1. Initial program 48.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot 2\right) \cdot \left(n \cdot U\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot 2} \cdot \sqrt{n \cdot U}} \]

      9. pow1/2 N/A

         \[\leadsto \sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot 2} \cdot \color{blue}{{\left(n \cdot U\right)}^{\frac{1}{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot 2} \cdot {\left(n \cdot U\right)}^{\frac{1}{2}}} \]

   4. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot 2} \cdot \sqrt{U \cdot n}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.3

         \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]

   7. Applied rewrites 42.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 35.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 6.5 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 6.5e-259)
   (sqrt (* (* (* n t) U) 2.0))
   (sqrt (* (* (* U t) n) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 6.5e-259) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = sqrt((((U * t) * n) * 2.0));
	}
	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 (u <= 6.5d-259) then
        tmp = sqrt((((n * t) * u) * 2.0d0))
    else
        tmp = sqrt((((u * t) * n) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 6.5e-259) {
		tmp = Math.sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((U * t) * n) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 6.5e-259:
		tmp = math.sqrt((((n * t) * U) * 2.0))
	else:
		tmp = math.sqrt((((U * t) * n) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 6.5e-259)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 6.5e-259)
		tmp = sqrt((((n * t) * U) * 2.0));
	else
		tmp = sqrt((((U * t) * n) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 6.5e-259], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 6.50000000000000045e-259

   1. Initial program 44.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 40.2

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 40.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   if 6.50000000000000045e-259 < U

   1. Initial program 48.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 29.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 29.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 33.4%

      \[\leadsto \sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 36.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* t (* (* U n) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((t * ((U * n) * 2.0)));
}

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((t * ((u * n) * 2.0d0)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((t * ((U * n) * 2.0)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((t * ((U * n) * 2.0)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(t * Float64(Float64(U * n) * 2.0)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((t * ((U * n) * 2.0)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(t * N[(N[(U * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 46.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   5. lower-*.f64 34.8

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 34.8%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 32.4%

   \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot 2\right)}} \]
8. Add Preprocessing

Alternative 28: 2.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{U} \cdot n\right) \cdot 2 \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (* (* (sqrt U) n) 2.0))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return (sqrt(U) * n) * 2.0;
}

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) * 2.0d0
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) * 2.0;
}

Python

def code(n, U, t, l, Om, U_42_):
	return (math.sqrt(U) * n) * 2.0

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(Float64(sqrt(U) * n) * 2.0)
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (sqrt(U) * n) * 2.0;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[(N[(N[Sqrt[U], $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 46.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

4. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, 2, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

5. Taylor expanded in n around inf

   \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

   4. lower--.f64 N/A

      \[\leadsto \sqrt{U \cdot \color{blue}{\left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{{Om}^{2}}\right)}} \cdot \left(n \cdot \sqrt{2}\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{U \cdot \left(2 - \frac{{\ell}^{2} \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   6. times-frac N/A

      \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om} \cdot \frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \sqrt{U \cdot \left(2 - \color{blue}{\frac{{\ell}^{2}}{Om}} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   9. unpow2 N/A

      \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{U \cdot \left(2 - \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \color{blue}{\frac{U - U*}{Om}}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   12. lower--.f64 N/A

       \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{\color{blue}{U - U*}}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \color{blue}{\left(n \cdot \sqrt{2}\right)} \]

   14. lower-sqrt.f64 11.1

       \[\leadsto \sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \color{blue}{\sqrt{2}}\right) \]

7. Applied rewrites 11.1%

   \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 - \frac{\ell \cdot \ell}{Om} \cdot \frac{U - U*}{Om}\right)} \cdot \left(n \cdot \sqrt{2}\right)} \]

8. Taylor expanded in l around 0

   \[\leadsto \sqrt{U} \cdot \color{blue}{\left(n \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 2.7%

    \[\leadsto \left(\sqrt{U} \cdot n\right) \cdot \color{blue}{2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024354 
(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*))))))