Toniolo and Linder, Equation (13)

Percentage Accurate: 50.4% → 64.6%

Time: 15.4s

Alternatives: 21

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% 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: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot t\_1\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) - \left(\left(U - U*\right) \cdot n\right) \cdot t\_1\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 10^{+154}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right)\right) \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 (/ (* l l) Om))) (* (* n t_1) (- U U*)))))))
   (if (<= t_2 0.0)
     (sqrt
      (*
       (* (- (fma (* (/ l Om) -2.0) l t) (* (* (- U U*) n) t_1)) (* 2.0 n))
       U))
     (if (<= t_2 1e+154)
       t_2
       (sqrt
        (*
         (* (* (* -2.0 U) l) (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n))
         l))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_1) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((((fma(((l / Om) * -2.0), l, t) - (((U - U_42_) * n) * t_1)) * (2.0 * n)) * U));
	} else if (t_2 <= 1e+154) {
		tmp = t_2;
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * ((fma((n / Om), (U - U_42_), 2.0) / Om) * n)) * l));
	}
	return tmp;
}

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] - N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+154], t$95$2, N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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 13.6

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

      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 13.6%

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

   6. Applied rewrites 46.2%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 96.3%

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

   if 1.00000000000000004e154 < (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 18.1%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 39.2%

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

   9. Applied rewrites 46.5%

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

Alternative 2: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 0.0)
     (sqrt (* (* (* U t) n) 2.0))
     (if (<= t_3 4e+302)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         (sqrt (* (* (* (fma (* (/ l Om) -2.0) l t) n) U) 2.0))
         (sqrt (* t_2 (/ (* U* (* (* l l) n)) (* Om 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 <= 0.0) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else if (t_3 <= 4e+302) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(((l / Om) * -2.0), l, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((t_2 * ((U_42_ * ((l * l) * n)) / (Om * 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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	elseif (t_3 <= 4e+302)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * 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, 0.0], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 4e+302], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 12.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 39.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.0000000000000003e302

   1. Initial program 96.3%

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

   3. 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 89.4

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

   5. Applied rewrites 89.4%

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

   if 4.0000000000000003e302 < (*.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 29.4%

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

   3. Taylor expanded in 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 16.4

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

   5. Applied rewrites 16.4%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 26.4

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

   8. Applied rewrites 26.4%

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

   10. Applied rewrites 35.9%

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

   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 Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 32.7%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 32.1%

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

Alternative 3: 53.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 0.0)
     (sqrt (* (* (* U t) n) 2.0))
     (if (<= t_3 4e+302)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         (sqrt (* (* (* (fma (* (/ l Om) -2.0) l t) n) U) 2.0))
         (sqrt (* 2.0 (/ (* (* U U*) (* (* l l) (* n n))) (* Om 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 <= 0.0) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else if (t_3 <= 4e+302) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(((l / Om) * -2.0), l, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((2.0 * (((U * U_42_) * ((l * l) * (n * n))) / (Om * 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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	elseif (t_3 <= 4e+302)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(U * U_42_) * Float64(Float64(l * l) * Float64(n * n))) / Float64(Om * 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, 0.0], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 4e+302], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 12.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 39.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.0000000000000003e302

   1. Initial program 96.3%

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

   3. 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 89.4

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

   5. Applied rewrites 89.4%

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

   if 4.0000000000000003e302 < (*.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 29.4%

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

   3. Taylor expanded in 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 16.4

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

   5. Applied rewrites 16.4%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 26.4

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

   8. Applied rewrites 26.4%

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

   10. Applied rewrites 35.9%

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

   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 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 28.8

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

   5. Applied rewrites 28.8%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 29.6%

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

Alternative 4: 53.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell \cdot n}{Om}\right)\right)}\\ \end{array} \end{array} \]

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 0.0)
     (sqrt (* (* (* U t) n) 2.0))
     (if (<= t_3 4e+302)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         (sqrt (* (* (* (fma (* (/ l Om) -2.0) l t) n) U) 2.0))
         (sqrt (* (* -2.0 U) (* l (* 2.0 (/ (* l n) 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 <= 0.0) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else if (t_3 <= 4e+302) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(((l / Om) * -2.0), l, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((-2.0 * U) * (l * (2.0 * ((l * n) / 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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	elseif (t_3 <= 4e+302)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(l * Float64(2.0 * Float64(Float64(l * n) / 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, 0.0], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 4e+302], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(l * N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 12.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 39.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.0000000000000003e302

   1. Initial program 96.3%

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

   3. 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 89.4

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

   5. Applied rewrites 89.4%

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

   if 4.0000000000000003e302 < (*.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 29.4%

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

   3. Taylor expanded in 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 16.4

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

   5. Applied rewrites 16.4%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 26.4

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

   8. Applied rewrites 26.4%

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

   10. Applied rewrites 35.9%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 53.6%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 21.8%

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

Alternative 5: 63.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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 13.6

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

      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 13.6%

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

   6. Applied rewrites 46.2%

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

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

   1. Initial program 66.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 72.9

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

      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 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 70.1

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

   6. Applied rewrites 70.1%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 52.5%

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

   9. Applied rewrites 61.5%

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

Alternative 6: 62.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.6%

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

   4. Applied rewrites 46.2%

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

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

   1. Initial program 66.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 72.9

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

      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 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 70.1

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

   6. Applied rewrites 70.1%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 52.5%

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

   9. Applied rewrites 61.5%

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

Alternative 7: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(-U\right) \cdot \mathsf{fma}\left(-2, t\_1, t\right)\right)\right) \cdot n}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right)\right) \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_3 2e-137)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_3 INFINITY)
       (sqrt
        (*
         t_2
         (fma
          (* -2.0 (/ l Om))
          l
          (- t (* (/ l Om) (* (/ l Om) (* (- U U*) n)))))))
       (sqrt
        (*
         (* (* (* -2.0 U) l) (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n))
         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 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 2e-137) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma((-2.0 * (l / Om)), l, (t - ((l / Om) * ((l / Om) * ((U - U_42_) * n)))))));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * ((fma((n / Om), (U - U_42_), 2.0) / Om) * n)) * 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 = sqrt(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 <= 2e-137)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(Float64(-2.0 * Float64(l / Om)), l, Float64(t - Float64(Float64(l / Om) * Float64(Float64(l / Om) * Float64(Float64(U - U_42_) * n)))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n)) * 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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-137], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(-2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l + N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * l), $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*))))) < 1.99999999999999996e-137

   1. Initial program 19.1%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 45.0%

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

   if 1.99999999999999996e-137 < (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 66.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 72.8

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

      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 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 69.8

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

   6. Applied rewrites 69.8%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 52.5%

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

   9. Applied rewrites 61.5%

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

Alternative 8: 58.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+154}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om} \cdot n\right)\right) \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* n t) U) 2.0))
     (if (<= t_3 1e+154)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt
        (*
         (* (* (* -2.0 U) l) (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n))
         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 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else if (t_3 <= 1e+154) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((((-2.0 * U) * l) * ((fma((n / Om), (U - U_42_), 2.0) / Om) * n)) * 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 = sqrt(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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	elseif (t_3 <= 1e+154)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n)) * 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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+154], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.6%

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

   3. Taylor expanded in 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 41.9

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

   5. Applied rewrites 41.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 96.3%

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

   3. 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 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.00000000000000004e154 < (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 18.1%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 39.2%

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

   9. Applied rewrites 46.5%

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

Alternative 9: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+154}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\ell \cdot \frac{\left(\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot n\right) \cdot \ell}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* n t) U) 2.0))
     (if (<= t_3 1e+154)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt
        (*
         (* -2.0 U)
         (* l (/ (* (* (fma (/ n Om) (- U U*) 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 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else if (t_3 <= 1e+154) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((-2.0 * U) * (l * (((fma((n / Om), (U - U_42_), 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 = sqrt(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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	elseif (t_3 <= 1e+154)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(l * Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+154], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(l * N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.6%

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

   3. Taylor expanded in 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 41.9

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

   5. Applied rewrites 41.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 96.3%

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

   3. 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 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.00000000000000004e154 < (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 18.1%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 39.2%

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

   9. Applied rewrites 43.4%

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

Alternative 10: 58.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+154}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* n t) U) 2.0))
     (if (<= t_3 1e+154)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt
        (*
         (* -2.0 U)
         (* l (* (* n l) (/ (fma (/ n Om) (- U U*) 2.0) 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 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else if (t_3 <= 1e+154) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((-2.0 * U) * (l * ((n * l) * (fma((n / Om), (U - U_42_), 2.0) / 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 = sqrt(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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	elseif (t_3 <= 1e+154)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(l * Float64(Float64(n * l) * Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / 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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+154], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(l * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.6%

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

   3. Taylor expanded in 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 41.9

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

   5. Applied rewrites 41.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 96.3%

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

   3. 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 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.00000000000000004e154 < (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 18.1%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 42.6%

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

Alternative 11: 56.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 1e-140)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_2 INFINITY)
       (* (sqrt (* (fma (* (/ l Om) -2.0) l t) (* n U))) (sqrt 2.0))
       (sqrt
        (*
         (* -2.0 U)
         (* l (* l (* (/ (fma (/ n Om) (- U U*) 2.0) Om) n)))))))))

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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e-140) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma(((l / Om) * -2.0), l, t) * (n * U))) * sqrt(2.0);
	} else {
		tmp = sqrt(((-2.0 * U) * (l * (l * ((fma((n / Om), (U - U_42_), 2.0) / Om) * n)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = 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_)))))
	tmp = 0.0
	if (t_2 <= 1e-140)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * Float64(n * U))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(l * Float64(l * Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om) * n)))));
	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[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]}, If[LessEqual[t$95$2, 1e-140], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(l * N[(l * N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999998e-141

   1. Initial program 15.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 42.6%

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

   if 9.9999999999999998e-141 < (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 66.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 51.9

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 65.7%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 52.5%

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

Alternative 12: 54.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 1e-140)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_2 INFINITY)
       (* (sqrt (* (fma (* (/ l Om) -2.0) l t) (* n U))) (sqrt 2.0))
       (sqrt (* (* l (* (* U l) (/ (* n (/ (* (- U U*) n) Om)) Om))) -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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e-140) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma(((l / Om) * -2.0), l, t) * (n * U))) * sqrt(2.0);
	} else {
		tmp = sqrt(((l * ((U * l) * ((n * (((U - U_42_) * n) / Om)) / Om))) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = 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_)))))
	tmp = 0.0
	if (t_2 <= 1e-140)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * Float64(n * U))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(U * l) * Float64(Float64(n * Float64(Float64(Float64(U - U_42_) * n) / Om)) / Om))) * -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[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]}, If[LessEqual[t$95$2, 1e-140], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(U * l), $MachinePrecision] * N[(N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -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*))))) < 9.9999999999999998e-141

   1. Initial program 15.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 42.6%

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

   if 9.9999999999999998e-141 < (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 66.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 51.9

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 65.7%

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

   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 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 26.8

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

   5. Applied rewrites 26.8%

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

   7. Applied rewrites 35.4%

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

Alternative 13: 53.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 1e-140)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_2 INFINITY)
       (* (sqrt (* (fma (* (/ l Om) -2.0) l t) (* n U))) (sqrt 2.0))
       (sqrt (* (* -2.0 U) (* (* (* l l) n) (* (/ U* (- Om)) (/ n 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e-140) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma(((l / Om) * -2.0), l, t) * (n * U))) * sqrt(2.0);
	} else {
		tmp = sqrt(((-2.0 * U) * (((l * l) * n) * ((U_42_ / -Om) * (n / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = 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_)))))
	tmp = 0.0
	if (t_2 <= 1e-140)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * Float64(n * U))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(Float64(Float64(l * l) * n) * Float64(Float64(U_42_ / Float64(-Om)) * Float64(n / 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[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]}, If[LessEqual[t$95$2, 1e-140], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(N[(U$42$ / (-Om)), $MachinePrecision] * N[(n / Om), $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*))))) < 9.9999999999999998e-141

   1. Initial program 15.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 42.6%

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

   if 9.9999999999999998e-141 < (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 66.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 51.9

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 65.7%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 33.1%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 10^{-140}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)\right) \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot \left(n \cdot U\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{U*}{-Om} \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_3 1e-140)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_3 INFINITY)
       (* (sqrt (* (fma (* (/ l Om) -2.0) l t) (* n U))) (sqrt 2.0))
       (sqrt (* t_2 (* (/ (* U* (* l l)) Om) (/ n 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 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 1e-140) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((fma(((l / Om) * -2.0), l, t) * (n * U))) * sqrt(2.0);
	} else {
		tmp = sqrt((t_2 * (((U_42_ * (l * l)) / Om) * (n / 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 = sqrt(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 <= 1e-140)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * Float64(n * U))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(Float64(U_42_ * Float64(l * l)) / Om) * Float64(n / 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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-140], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $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*))))) < 9.9999999999999998e-141

   1. Initial program 15.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 42.6%

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

   if 9.9999999999999998e-141 < (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 66.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 51.9

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 65.7%

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

   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 U* around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 32.5

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

   5. Applied rewrites 32.5%

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

Alternative 15: 54.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{-280}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(-U\right) \cdot \mathsf{fma}\left(-2, t\_1, t\right)\right)\right) \cdot n}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot \left(n \cdot U\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (*
          (* (* 2.0 n) U)
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 1e-280)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_2 INFINITY)
       (* (sqrt (* (fma (* (/ l Om) -2.0) l t) (* n U))) (sqrt 2.0))
       (sqrt
        (*
         (* -2.0 U)
         (* (* (* l l) n) (/ (fma 2.0 Om (* n (- U U*))) (* Om Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 1e-280) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma(((l / Om) * -2.0), l, t) * (n * U))) * sqrt(2.0);
	} else {
		tmp = sqrt(((-2.0 * U) * (((l * l) * n) * (fma(2.0, Om, (n * (U - U_42_))) / (Om * Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = 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_))))
	tmp = 0.0
	if (t_2 <= 1e-280)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * Float64(n * U))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(Float64(Float64(l * l) * n) * Float64(fma(2.0, Om, Float64(n * Float64(U - U_42_))) / Float64(Om * 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[(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]}, If[LessEqual[t$95$2, 1e-280], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(N[(2.0 * Om + N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $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*)))) < 9.9999999999999996e-281

   1. Initial program 14.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 40.1%

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

   if 9.9999999999999996e-281 < (*.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 66.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 51.9

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 65.7%

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

   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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 36.3%

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

Alternative 16: 53.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 10^{-280}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(-U\right) \cdot \mathsf{fma}\left(-2, t\_1, t\right)\right)\right) \cdot n}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot \left(n \cdot U\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 1e-280)
     (sqrt (* (* -2.0 (* (- U) (fma -2.0 t_1 t))) n))
     (if (<= t_3 INFINITY)
       (* (sqrt (* (fma (* (/ l Om) -2.0) l t) (* n U))) (sqrt 2.0))
       (sqrt (* t_2 (/ (* U* (* (* l l) n)) (* Om 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 <= 1e-280) {
		tmp = sqrt(((-2.0 * (-U * fma(-2.0, t_1, t))) * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((fma(((l / Om) * -2.0), l, t) * (n * U))) * sqrt(2.0);
	} else {
		tmp = sqrt((t_2 * ((U_42_ * ((l * l) * n)) / (Om * 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 <= 1e-280)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(-U) * fma(-2.0, t_1, t))) * n));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * Float64(n * U))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * 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, 1e-280], N[Sqrt[N[(N[(-2.0 * N[((-U) * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $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*)))) < 9.9999999999999996e-281

   1. Initial program 14.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 40.1%

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

   if 9.9999999999999996e-281 < (*.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 66.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 51.9

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 65.7%

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

   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 Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 32.7%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 32.1%

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

Alternative 17: 42.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot -2, \ell, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell \cdot n}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.6e-64)
   (sqrt (* (* (* U t) n) 2.0))
   (if (<= l 3e+161)
     (sqrt (* (* (* (fma (* (/ l Om) -2.0) l t) n) U) 2.0))
     (sqrt (* (* -2.0 U) (* l (* 2.0 (/ (* l n) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.6e-64) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else if (l <= 3e+161) {
		tmp = sqrt((((fma(((l / Om) * -2.0), l, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((-2.0 * U) * (l * (2.0 * ((l * n) / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.6e-64)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	elseif (l <= 3e+161)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(Float64(l / Om) * -2.0), l, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(l * Float64(2.0 * Float64(Float64(l * n) / Om)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.6e-64], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3e+161], N[Sqrt[N[(N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] * l + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(l * N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.59999999999999988e-64

   1. Initial program 48.3%

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

   3. 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 39.9

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

   5. Applied rewrites 39.9%

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

   7. Applied rewrites 44.7%

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

   if 1.59999999999999988e-64 < l < 3.00000000000000011e161

   1. Initial program 66.3%

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

   3. 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 37.0

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

   5. Applied rewrites 37.0%

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

   6. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 57.3

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

   8. Applied rewrites 57.3%

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

   10. Applied rewrites 62.2%

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

   if 3.00000000000000011e161 < l

   1. Initial program 11.3%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 28.5

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

   5. Applied rewrites 28.5%

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 40.8%

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

Alternative 18: 40.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell \cdot n}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e+64)
   (sqrt (* (* (* U t) n) 2.0))
   (sqrt (* (* -2.0 U) (* l (* 2.0 (/ (* l n) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+64) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = sqrt(((-2.0 * U) * (l * (2.0 * ((l * n) / Om)))));
	}
	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 (l <= 1.15d+64) then
        tmp = sqrt((((u * t) * n) * 2.0d0))
    else
        tmp = sqrt((((-2.0d0) * u) * (l * (2.0d0 * ((l * n) / om)))))
    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 (l <= 1.15e+64) {
		tmp = Math.sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = Math.sqrt(((-2.0 * U) * (l * (2.0 * ((l * n) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e+64:
		tmp = math.sqrt((((U * t) * n) * 2.0))
	else:
		tmp = math.sqrt(((-2.0 * U) * (l * (2.0 * ((l * n) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e+64)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(l * Float64(2.0 * Float64(Float64(l * n) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e+64)
		tmp = sqrt((((U * t) * n) * 2.0));
	else
		tmp = sqrt(((-2.0 * U) * (l * (2.0 * ((l * n) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e+64], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(l * N[(2.0 * N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e64

   1. Initial program 51.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 44.5%

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

   if 1.15e64 < l

   1. Initial program 23.5%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 35.5

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

   5. Applied rewrites 35.5%

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 44.4%

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

Alternative 19: 37.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.2e+58)
   (sqrt (* (* (* U t) n) 2.0))
   (sqrt (* (* -2.0 U) (* 2.0 (/ (* (* l l) n) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.2e+58) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))));
	}
	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 (l <= 7.2d+58) then
        tmp = sqrt((((u * t) * n) * 2.0d0))
    else
        tmp = sqrt((((-2.0d0) * u) * (2.0d0 * (((l * l) * n) / om))))
    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 (l <= 7.2e+58) {
		tmp = Math.sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = Math.sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7.2e+58:
		tmp = math.sqrt((((U * t) * n) * 2.0))
	else:
		tmp = math.sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.2e+58)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(2.0 * Float64(Float64(Float64(l * l) * n) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 7.2e+58)
		tmp = sqrt((((U * t) * n) * 2.0));
	else
		tmp = sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.2e+58], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.19999999999999993e58

   1. Initial program 51.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 44.5%

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

   if 7.19999999999999993e58 < l

   1. Initial program 23.5%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 22.6%

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

Alternative 20: 36.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 4 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot t\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U + U\right) \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 4e-91) (sqrt (* (* (* U t) n) 2.0)) (sqrt (* t (* (+ U U) n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 4e-91) {
		tmp = sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = sqrt((t * ((U + 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 (u <= 4d-91) then
        tmp = sqrt((((u * t) * n) * 2.0d0))
    else
        tmp = sqrt((t * ((u + 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 (U <= 4e-91) {
		tmp = Math.sqrt((((U * t) * n) * 2.0));
	} else {
		tmp = Math.sqrt((t * ((U + U) * n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 4e-91:
		tmp = math.sqrt((((U * t) * n) * 2.0))
	else:
		tmp = math.sqrt((t * ((U + U) * n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 4e-91)
		tmp = sqrt(Float64(Float64(Float64(U * t) * n) * 2.0));
	else
		tmp = sqrt(Float64(t * Float64(Float64(U + U) * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 4e-91)
		tmp = sqrt((((U * t) * n) * 2.0));
	else
		tmp = sqrt((t * ((U + U) * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 4e-91], N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 4.00000000000000009e-91

   1. Initial program 42.3%

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

   3. 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 33.2

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

   5. Applied rewrites 33.2%

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

   7. Applied rewrites 38.9%

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

   if 4.00000000000000009e-91 < U

   1. Initial program 58.0%

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

   3. Taylor expanded in 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 39.9

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

   5. Applied rewrites 39.9%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 46.9%

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

Alternative 21: 35.7% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

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 + u) * n)))
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 + U) * n)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
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 35.1

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

5. Applied rewrites 35.1%

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

7. Applied rewrites 35.9%

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

9. Applied rewrites 35.9%

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

Reproduce

herbie shell --seed 2024360 
(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*))))))