Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 61.6%

Time: 9.8s

Alternatives: 16

Speedup: 1.4×

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: 49.9% 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: 61.6% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        INFINITY)
     (sqrt
      (*
       t_1
       (+
        (fma -2.0 (* l_m (/ l_m Om)) t)
        (* (- (* (* (/ l_m Om) (/ l_m Om)) n)) (- U U*)))))
     (*
      (sqrt
       (*
        U
        (* n (- (* -1.0 (/ (* n (- U U*)) (* Om Om))) (* 2.0 (/ 1.0 Om))))))
      (* l_m (sqrt 2.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 63.2%

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

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

   1. Initial program 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. 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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 2.8%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   6. Applied rewrites 53.0%

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

Alternative 2: 60.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 1.8 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(-\left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;l\_m \leq 2.3 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(-\frac{\mathsf{fma}\left(l\_m \cdot l\_m, \frac{\left(U - U*\right) \cdot n}{Om}, \left(l\_m \cdot l\_m\right) \cdot 2\right)}{Om}\right) + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(-1 \cdot \frac{n \cdot \left(U - U*\right)}{Om \cdot Om} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 1.8e-134)
     (sqrt (* t_1 (+ t (* (- (* (* (/ l_m Om) (/ l_m Om)) n)) (- U U*)))))
     (if (<= l_m 2.3e+138)
       (sqrt
        (*
         t_1
         (+
          (-
           (/ (fma (* l_m l_m) (/ (* (- U U*) n) Om) (* (* l_m l_m) 2.0)) Om))
          t)))
       (*
        (sqrt
         (*
          U
          (* n (- (* -1.0 (/ (* n (- U U*)) (* Om Om))) (* 2.0 (/ 1.0 Om))))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 1.8e-134) {
		tmp = sqrt((t_1 * (t + (-(((l_m / Om) * (l_m / Om)) * n) * (U - U_42_)))));
	} else if (l_m <= 2.3e+138) {
		tmp = sqrt((t_1 * (-(fma((l_m * l_m), (((U - U_42_) * n) / Om), ((l_m * l_m) * 2.0)) / Om) + t)));
	} else {
		tmp = sqrt((U * (n * ((-1.0 * ((n * (U - U_42_)) / (Om * Om))) - (2.0 * (1.0 / Om)))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 1.8e-134)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(-Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n)) * Float64(U - U_42_)))));
	elseif (l_m <= 2.3e+138)
		tmp = sqrt(Float64(t_1 * Float64(Float64(-Float64(fma(Float64(l_m * l_m), Float64(Float64(Float64(U - U_42_) * n) / Om), Float64(Float64(l_m * l_m) * 2.0)) / Om)) + t)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(-1.0 * Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om))) - Float64(2.0 * Float64(1.0 / Om)))))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 1.8e-134], N[Sqrt[N[(t$95$1 * N[(t + N[((-N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.3e+138], N[Sqrt[N[(t$95$1 * N[((-N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(-1.0 * N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.79999999999999995e-134

   1. Initial program 64.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 64.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 64.5%

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

   if 1.79999999999999995e-134 < l < 2.30000000000000008e138

   1. Initial program 56.5%

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

   2. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 58.2%

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

   if 2.30000000000000008e138 < l

   1. Initial program 18.3%

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

      1. lift--.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 33.2%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   6. Applied rewrites 57.1%

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

Alternative 3: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(-\left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;l\_m \leq 8.8 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(-\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(-1 \cdot \frac{n \cdot \left(U - U*\right)}{Om \cdot Om} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 8e+14)
     (sqrt (* t_1 (+ t (* (- (* (* (/ l_m Om) (/ l_m Om)) n)) (- U U*)))))
     (if (<= l_m 8.8e+144)
       (sqrt
        (*
         t_1
         (+ (- (* (* l_m l_m) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)))) t)))
       (*
        (sqrt
         (*
          U
          (* n (- (* -1.0 (/ (* n (- U U*)) (* Om Om))) (* 2.0 (/ 1.0 Om))))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 8e+14) {
		tmp = sqrt((t_1 * (t + (-(((l_m / Om) * (l_m / Om)) * n) * (U - U_42_)))));
	} else if (l_m <= 8.8e+144) {
		tmp = sqrt((t_1 * (-((l_m * l_m) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) + t)));
	} else {
		tmp = sqrt((U * (n * ((-1.0 * ((n * (U - U_42_)) / (Om * Om))) - (2.0 * (1.0 / Om)))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 8e+14)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(-Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n)) * Float64(U - U_42_)))));
	elseif (l_m <= 8.8e+144)
		tmp = sqrt(Float64(t_1 * Float64(Float64(-Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om)))) + t)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(-1.0 * Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om))) - Float64(2.0 * Float64(1.0 / Om)))))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 8e+14], N[Sqrt[N[(t$95$1 * N[(t + N[((-N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 8.8e+144], N[Sqrt[N[(t$95$1 * N[((-N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(-1.0 * N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8e14

   1. Initial program 62.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 60.4%

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

   if 8e14 < l < 8.79999999999999952e144

   1. Initial program 53.7%

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

   2. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 58.6%

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

   if 8.79999999999999952e144 < l

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 32.5%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   6. Applied rewrites 57.5%

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

Alternative 4: 54.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{if}\;Om \leq -8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 2.75 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(-\left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))))
   (if (<= Om -8e-18)
     t_1
     (if (<= Om 2.75e+158)
       (sqrt
        (*
         (* (* 2.0 n) U)
         (+ t (* (- (* (* (/ l_m Om) (/ l_m Om)) n)) (- U U*)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	double tmp;
	if (Om <= -8e-18) {
		tmp = t_1;
	} else if (Om <= 2.75e+158) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-(((l_m / Om) * (l_m / Om)) * n) * (U - U_42_)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0))
	tmp = 0.0
	if (Om <= -8e-18)
		tmp = t_1;
	elseif (Om <= 2.75e+158)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(-Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n)) * Float64(U - U_42_)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -8e-18], t$95$1, If[LessEqual[Om, 2.75e+158], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[((-N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -8.0000000000000006e-18 or 2.7499999999999999e158 < Om

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 57.8

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

   4. Applied rewrites 57.8%

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

   if -8.0000000000000006e-18 < Om < 2.7499999999999999e158

   1. Initial program 47.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 48.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 51.2%

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

Alternative 5: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{if}\;Om \leq -8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \frac{l\_m \cdot l\_m}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))))
   (if (<= Om -8e-18)
     t_1
     (if (<= Om 7.5e+153)
       (sqrt
        (* (* (* 2.0 n) U) (- t (* (* n (/ (* l_m l_m) (* Om Om))) (- U U*)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	double tmp;
	if (Om <= -8e-18) {
		tmp = t_1;
	} else if (Om <= 7.5e+153) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * ((l_m * l_m) / (Om * Om))) * (U - U_42_)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0))
	tmp = 0.0
	if (Om <= -8e-18)
		tmp = t_1;
	elseif (Om <= 7.5e+153)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l_m * l_m) / Float64(Om * Om))) * Float64(U - U_42_)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -8e-18], t$95$1, If[LessEqual[Om, 7.5e+153], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -8.0000000000000006e-18 or 7.50000000000000065e153 < Om

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 57.9

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

   4. Applied rewrites 57.9%

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

   if -8.0000000000000006e-18 < Om < 7.50000000000000065e153

   1. Initial program 47.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 51.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 42.9

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

   6. Applied rewrites 42.9%

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

Alternative 6: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{if}\;Om \leq -8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))))
   (if (<= Om -8e-18)
     t_1
     (if (<= Om 7e+153)
       (sqrt (* (* (* 2.0 n) U) (+ t (/ (* U* (* (* l_m l_m) n)) (* Om Om)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	double tmp;
	if (Om <= -8e-18) {
		tmp = t_1;
	} else if (Om <= 7e+153) {
		tmp = sqrt((((2.0 * n) * U) * (t + ((U_42_ * ((l_m * l_m) * n)) / (Om * Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0))
	tmp = 0.0
	if (Om <= -8e-18)
		tmp = t_1;
	elseif (Om <= 7e+153)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(Om * Om)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -8e-18], t$95$1, If[LessEqual[Om, 7e+153], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -8.0000000000000006e-18 or 6.9999999999999998e153 < Om

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 57.9

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

   4. Applied rewrites 57.9%

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

   if -8.0000000000000006e-18 < Om < 6.9999999999999998e153

   1. Initial program 47.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 48.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in U around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 42.7

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

   9. Applied rewrites 42.7%

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

Alternative 7: 50.3% accurate, 0.6× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        INFINITY)
     (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
     (sqrt (* (* (/ (* U* U) Om) (/ (* (* n l_m) (* n l_m)) Om)) 2.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 55.1

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

   4. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 23.6

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

   4. Applied rewrites 23.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 26.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unswap-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 31.8

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

   6. Applied rewrites 31.8%

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

Alternative 8: 49.9% accurate, 0.6× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        INFINITY)
     (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
     (sqrt (* (/ (* (* U* U) (* (* n l_m) (* n l_m))) (* Om Om)) 2.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 55.1

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

   4. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 23.6

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

   4. Applied rewrites 23.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 28.1

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

   6. Applied rewrites 28.1%

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

Alternative 9: 49.8% accurate, 0.6× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        INFINITY)
     (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
     (sqrt (* (/ (* U (* U* (* (* l_m l_m) (* n n)))) (* Om Om)) 2.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 55.1

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

   4. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

Alternative 10: 49.7% accurate, 0.7× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        INFINITY)
     (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
     (* -1.0 (* (/ (* l_m (* n (sqrt 2.0))) Om) (sqrt (* U U*)))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 55.1

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

   4. Applied rewrites 55.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 6.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 7.9

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

   6. Applied rewrites 7.9%

      \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

   7. Taylor expanded in U* around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 20.8

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

   9. Applied rewrites 20.8%

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

Alternative 11: 49.6% accurate, 0.7× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
      INFINITY)
   (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
   (* -1.0 (* (/ (* l_m (* n (sqrt 2.0))) Om) (sqrt (* U U*))))))

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], Infinity], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(-1.0 * N[(N[(N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 55.5

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

   4. Applied rewrites 55.5%

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

   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. Taylor expanded in t around inf

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

   4. Applied rewrites 6.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 7.9

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

   6. Applied rewrites 7.9%

      \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

   7. Taylor expanded in U* around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 20.8

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

   9. Applied rewrites 20.8%

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

Alternative 12: 41.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{l\_m \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
      1e+152)
   (sqrt (* (* (* t n) U) 2.0))
   (* -1.0 (* (/ (* l_m (* n (sqrt 2.0))) Om) (sqrt (* U U*))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 1e+152) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = -1.0 * (((l_m * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_)));
	}
	return tmp;
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42))))) <= 1d+152) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = (-1.0d0) * (((l_m * (n * sqrt(2.0d0))) / om) * sqrt((u * u_42)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 1e+152) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = -1.0 * (((l_m * (n * Math.sqrt(2.0))) / Om) * Math.sqrt((U * U_42_)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 1e+152:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = -1.0 * (((l_m * (n * math.sqrt(2.0))) / Om) * math.sqrt((U * U_42_)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 1e+152)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(l_m * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U * U_42_))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_))))) <= 1e+152)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = -1.0 * (((l_m * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+152], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(-1.0 * N[(N[(N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e152

   1. Initial program 76.3%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.5

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

   4. Applied rewrites 59.5%

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

   if 1e152 < (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 20.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 9.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 12.4

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

   6. Applied rewrites 12.4%

      \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

   7. Taylor expanded in U* around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 22.0

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

   9. Applied rewrites 22.0%

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

Alternative 13: 41.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot l\_m}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
      1e+152)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l_m) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 1e+152) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l_m) / Om);
	}
	return tmp;
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42))))) <= 1d+152) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((u_42 * u)) * (((sqrt(2.0d0) * n) * l_m) / om)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 1e+152) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l_m) / Om);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 1e+152:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l_m) / Om)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 1e+152)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l_m) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_))))) <= 1e+152)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l_m) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+152], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e152

   1. Initial program 76.3%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.5

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

   4. Applied rewrites 59.5%

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

   if 1e152 < (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 20.7%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 21.5

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

   4. Applied rewrites 21.5%

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

Alternative 14: 40.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.7e+82)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* -4.0 (/ (* U (* (* l_m l_m) n)) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e+82) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)));
	}
	return tmp;
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.7d+82) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((-4.0d0) * ((u * ((l_m * l_m) * n)) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e+82) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.7e+82:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.7e+82)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(Float64(l_m * l_m) * n)) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.7e+82)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.7e+82], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.6999999999999999e82

   1. Initial program 61.0%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 48.7

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

   4. Applied rewrites 48.7%

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

   if 2.6999999999999999e82 < l

   1. Initial program 26.7%

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

   2. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 24.8

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

   4. Applied rewrites 24.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      4. pow2 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}} \]

      6. lift-*.f64 22.8

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}} \]

   7. Applied rewrites 22.8%

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

Alternative 15: 39.4% accurate, 3.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= t -1e-309) (sqrt (* t_1 t)) (* (sqrt t_1) (sqrt t)))))

C

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

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (t <= (-1d-309)) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt(t_1) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (t <= -1e-309) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if t <= -1e-309:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt(t_1) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 50.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 36.4%

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

   if -1.000000000000002e-309 < t

   1. Initial program 49.5%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 35.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 42.4

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

   6. Applied rewrites 42.4%

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

Alternative 16: 36.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* t n) U) 2.0)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((((t * n) * U) * 2.0));
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((t * n) * u) * 2.0d0))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((((t * n) * U) * 2.0));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((t * n) * U) * 2.0))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(t * n) * U) * 2.0))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((t * n) * U) * 2.0));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 36.7

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

4. Applied rewrites 36.7%

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

Reproduce

herbie shell --seed 2025120 
(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*))))))