Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 63.1%

Time: 10.4s

Alternatives: 11

Speedup: 1.9×

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: 49.6% 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: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot n, \frac{l\_m}{Om} \cdot \left(U* - U\right), \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right)}\\ \mathbf{elif}\;l\_m \leq 5.6 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot \left(t - \frac{\mathsf{fma}\left(2 \cdot l\_m, l\_m, \left(U - U*\right) \cdot \left(\left(\frac{l\_m}{Om} \cdot l\_m\right) \cdot n\right)\right)}{Om}\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.1e-56)
   (sqrt
    (*
     (* (* 2.0 n) U)
     (fma
      (* (/ l_m Om) n)
      (* (/ l_m Om) (- U* U))
      (fma -2.0 (/ (* l_m l_m) Om) t))))
   (if (<= l_m 5.6e+138)
     (sqrt
      (*
       (*
        (+ n n)
        (-
         t
         (/ (fma (* 2.0 l_m) l_m (* (- U U*) (* (* (/ l_m Om) l_m) n))) Om)))
       U))
     (*
      l_m
      (sqrt
       (*
        -2.0
        (* U (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow 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 tmp;
	if (l_m <= 3.1e-56) {
		tmp = sqrt((((2.0 * n) * U) * fma(((l_m / Om) * n), ((l_m / Om) * (U_42_ - U)), fma(-2.0, ((l_m * l_m) / Om), t))));
	} else if (l_m <= 5.6e+138) {
		tmp = sqrt((((n + n) * (t - (fma((2.0 * l_m), l_m, ((U - U_42_) * (((l_m / Om) * l_m) * n))) / Om))) * U));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.1e-56)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * fma(Float64(Float64(l_m / Om) * n), Float64(Float64(l_m / Om) * Float64(U_42_ - U)), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))));
	elseif (l_m <= 5.6e+138)
		tmp = sqrt(Float64(Float64(Float64(n + n) * Float64(t - Float64(fma(Float64(2.0 * l_m), l_m, Float64(Float64(U - U_42_) * Float64(Float64(Float64(l_m / Om) * l_m) * n))) / Om))) * U));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.1e-56], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 5.6e+138], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * N[(t - N[(N[(N[(2.0 * l$95$m), $MachinePrecision] * l$95$m + N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.09999999999999987e-56

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.0%

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

   if 3.09999999999999987e-56 < l < 5.6000000000000002e138

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 53.8

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 53.9%

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

   if 5.6000000000000002e138 < l

   1. Initial program 49.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.5

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

   4. Applied rewrites 27.5%

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

Alternative 2: 61.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{-314}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot \left(t - \frac{\mathsf{fma}\left(\left(U - U*\right) \cdot \frac{l\_m}{Om}, l\_m \cdot n, \left(l\_m + l\_m\right) \cdot l\_m\right)}{Om}\right)\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) - \left(\frac{l\_m}{Om} \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 1e-314)
     (sqrt
      (*
       (*
        (+ n n)
        (-
         t
         (/ (fma (* (- U U*) (/ l_m Om)) (* l_m n) (* (+ l_m l_m) l_m)) Om)))
       U))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         t_1
         (-
          (fma (/ l_m Om) (* -2.0 l_m) t)
          (* (* (/ l_m Om) (* (/ l_m Om) n)) (- U U*)))))
       (sqrt (fabs (* (* (fma -2.0 (* (/ l_m Om) l_m) t) U) (+ n n))))))))

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 t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 1e-314) {
		tmp = sqrt((((n + n) * (t - (fma(((U - U_42_) * (l_m / Om)), (l_m * n), ((l_m + l_m) * l_m)) / Om))) * U));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (fma((l_m / Om), (-2.0 * l_m), t) - (((l_m / Om) * ((l_m / Om) * n)) * (U - U_42_)))));
	} else {
		tmp = sqrt(fabs(((fma(-2.0, ((l_m / Om) * l_m), t) * U) * (n + n))));
	}
	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)
	t_2 = 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_))))
	tmp = 0.0
	if (t_2 <= 1e-314)
		tmp = sqrt(Float64(Float64(Float64(n + n) * Float64(t - Float64(fma(Float64(Float64(U - U_42_) * Float64(l_m / Om)), Float64(l_m * n), Float64(Float64(l_m + l_m) * l_m)) / Om))) * U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) - Float64(Float64(Float64(l_m / Om) * Float64(Float64(l_m / Om) * n)) * Float64(U - U_42_)))));
	else
		tmp = sqrt(abs(Float64(Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U) * Float64(n + n))));
	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]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 1e-314], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision] + N[(N[(l$95$m + l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] - N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.9999999996e-315

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 53.8

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 53.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.6

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 50.6

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

   9. Applied rewrites 50.6%

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

   if 9.9999999996e-315 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 53.8

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

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

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   10. Applied rewrites 52.2%

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

Alternative 3: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot n, \frac{l\_m}{Om} \cdot \left(U* - U\right), \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right)}\\ \mathbf{elif}\;l\_m \leq 6.2 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot \left(t - \frac{\mathsf{fma}\left(\left(U - U*\right) \cdot \frac{l\_m}{Om}, l\_m \cdot n, \left(l\_m + l\_m\right) \cdot l\_m\right)}{Om}\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.85e-68)
   (sqrt
    (*
     (* (* 2.0 n) U)
     (fma
      (* (/ l_m Om) n)
      (* (/ l_m Om) (- U* U))
      (fma -2.0 (/ (* l_m l_m) Om) t))))
   (if (<= l_m 6.2e+52)
     (sqrt
      (*
       (*
        (+ n n)
        (-
         t
         (/ (fma (* (- U U*) (/ l_m Om)) (* l_m n) (* (+ l_m l_m) l_m)) Om)))
       U))
     (sqrt (fabs (* (* (fma -2.0 (* (/ l_m Om) l_m) t) U) (+ n n)))))))

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 <= 2.85e-68) {
		tmp = sqrt((((2.0 * n) * U) * fma(((l_m / Om) * n), ((l_m / Om) * (U_42_ - U)), fma(-2.0, ((l_m * l_m) / Om), t))));
	} else if (l_m <= 6.2e+52) {
		tmp = sqrt((((n + n) * (t - (fma(((U - U_42_) * (l_m / Om)), (l_m * n), ((l_m + l_m) * l_m)) / Om))) * U));
	} else {
		tmp = sqrt(fabs(((fma(-2.0, ((l_m / Om) * l_m), t) * U) * (n + n))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.85e-68)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * fma(Float64(Float64(l_m / Om) * n), Float64(Float64(l_m / Om) * Float64(U_42_ - U)), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))));
	elseif (l_m <= 6.2e+52)
		tmp = sqrt(Float64(Float64(Float64(n + n) * Float64(t - Float64(fma(Float64(Float64(U - U_42_) * Float64(l_m / Om)), Float64(l_m * n), Float64(Float64(l_m + l_m) * l_m)) / Om))) * U));
	else
		tmp = sqrt(abs(Float64(Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U) * Float64(n + n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.85e-68], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 6.2e+52], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision] + N[(N[(l$95$m + l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.8500000000000001e-68

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.0%

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

   if 2.8500000000000001e-68 < l < 6.2e52

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 53.8

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 53.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.6

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 50.6

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

   9. Applied rewrites 50.6%

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

   if 6.2e52 < l

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   10. Applied rewrites 52.2%

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

Alternative 4: 56.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 6.2 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot \left(t - \frac{\mathsf{fma}\left(\left(U - U*\right) \cdot \frac{l\_m}{Om}, l\_m \cdot n, \left(l\_m + l\_m\right) \cdot l\_m\right)}{Om}\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.2e+52)
   (sqrt
    (*
     (*
      (+ n n)
      (- t (/ (fma (* (- U U*) (/ l_m Om)) (* l_m n) (* (+ l_m l_m) l_m)) Om)))
     U))
   (sqrt (fabs (* (* (fma -2.0 (* (/ l_m Om) l_m) t) U) (+ n n))))))

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 <= 6.2e+52) {
		tmp = sqrt((((n + n) * (t - (fma(((U - U_42_) * (l_m / Om)), (l_m * n), ((l_m + l_m) * l_m)) / Om))) * U));
	} else {
		tmp = sqrt(fabs(((fma(-2.0, ((l_m / Om) * l_m), t) * U) * (n + n))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.2e+52)
		tmp = sqrt(Float64(Float64(Float64(n + n) * Float64(t - Float64(fma(Float64(Float64(U - U_42_) * Float64(l_m / Om)), Float64(l_m * n), Float64(Float64(l_m + l_m) * l_m)) / Om))) * U));
	else
		tmp = sqrt(abs(Float64(Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U) * Float64(n + n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 6.2e+52], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision] + N[(N[(l$95$m + l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.2e52

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 53.8

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 53.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.6

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 50.6

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

   9. Applied rewrites 50.6%

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

   if 6.2e52 < l

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   10. Applied rewrites 52.2%

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

Alternative 5: 53.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.8 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 6.3 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* \cdot n, \frac{l\_m}{Om \cdot Om} \cdot l\_m, \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.8e-107)
   (sqrt (fabs (* t (* U (+ n n)))))
   (if (<= l_m 6.3e+29)
     (sqrt
      (*
       (+ n n)
       (*
        (fma
         (* U* n)
         (* (/ l_m (* Om Om)) l_m)
         (fma -2.0 (/ (* l_m l_m) Om) t))
        U)))
     (sqrt (fabs (* (* (fma -2.0 (* (/ l_m Om) l_m) t) U) (+ n n)))))))

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 <= 2.8e-107) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else if (l_m <= 6.3e+29) {
		tmp = sqrt(((n + n) * (fma((U_42_ * n), ((l_m / (Om * Om)) * l_m), fma(-2.0, ((l_m * l_m) / Om), t)) * U)));
	} else {
		tmp = sqrt(fabs(((fma(-2.0, ((l_m / Om) * l_m), t) * U) * (n + n))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.8e-107)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	elseif (l_m <= 6.3e+29)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(U_42_ * n), Float64(Float64(l_m / Float64(Om * Om)) * l_m), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)) * U)));
	else
		tmp = sqrt(abs(Float64(Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U) * Float64(n + n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.8e-107], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 6.3e+29], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(U$42$ * n), $MachinePrecision] * N[(N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.7999999999999999e-107

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.8%

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

   if 2.7999999999999999e-107 < l < 6.2999999999999998e29

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

   4. Taylor expanded in U around 0

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

   6. Applied rewrites 42.6%

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

   if 6.2999999999999998e29 < l

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   10. Applied rewrites 52.2%

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

Alternative 6: 52.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.1 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right) \cdot \left(n + n\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.1e-68)
   (sqrt (fabs (* t (* U (+ n n)))))
   (sqrt (fabs (* (* (fma -2.0 (* (/ l_m Om) l_m) t) U) (+ n n))))))

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 <= 2.1e-68) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = sqrt(fabs(((fma(-2.0, ((l_m / Om) * l_m), t) * U) * (n + n))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.1e-68)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = sqrt(abs(Float64(Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U) * Float64(n + n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.1e-68], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.10000000000000008e-68

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.8%

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

   if 2.10000000000000008e-68 < l

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   10. Applied rewrites 52.2%

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

Alternative 7: 52.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(-1 \cdot \frac{l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 5e-129)
     (sqrt (* (+ n n) (* (fma -2.0 (* (/ l_m Om) l_m) t) U)))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma (/ l_m Om) (* -2.0 l_m) t)))
       (* n (* -1.0 (/ (* l_m (sqrt (* -2.0 (* U (- U U*))))) Om)))))))

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 t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 5e-129) {
		tmp = sqrt(((n + n) * (fma(-2.0, ((l_m / Om) * l_m), t) * U)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma((l_m / Om), (-2.0 * l_m), t)));
	} else {
		tmp = n * (-1.0 * ((l_m * sqrt((-2.0 * (U * (U - U_42_))))) / Om));
	}
	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)
	t_2 = 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_)))))
	tmp = 0.0
	if (t_2 <= 5e-129)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), Float64(-2.0 * l_m), t)));
	else
		tmp = Float64(n * Float64(-1.0 * Float64(Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(U - U_42_))))) / Om)));
	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]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 5e-129], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[(-1.0 * N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000027e-129

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 46.6%

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

   if 5.00000000000000027e-129 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 49.6%

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

   3. Applied rewrites 47.0%

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

   4. Taylor expanded in n around 0

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

   6. Applied rewrites 47.0%

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

   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 49.6%

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

   2. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 9.2

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

   4. Applied rewrites 9.2%

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

   5. Taylor expanded in Om around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 10.9

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

   7. Applied rewrites 10.9%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 13.8

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

   10. Applied rewrites 13.8%

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

Alternative 8: 47.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.1e-62)
   (sqrt (fabs (* t (* U (+ n n)))))
   (sqrt (* (+ n n) (* (fma -2.0 (* (/ l_m Om) l_m) t) 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 (l_m <= 3.1e-62) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = sqrt(((n + n) * (fma(-2.0, ((l_m / Om) * l_m), t) * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.1e-62)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(-2.0, Float64(Float64(l_m / Om) * l_m), t) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.1e-62], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.0999999999999999e-62

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.8%

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

   if 3.0999999999999999e-62 < l

   1. Initial program 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.3

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

   3. Applied rewrites 41.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 43.1

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

   8. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 46.6%

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

Alternative 9: 40.3% accurate, 0.8× speedup

Math

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

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*)))))
      0.0)
   (sqrt (* (+ U U) (* t n)))
   (sqrt (fabs (* t (* U (+ n n)))))))

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_))))) <= 0.0) {
		tmp = sqrt(((U + U) * (t * n)));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	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))))) <= 0.0d0) then
        tmp = sqrt(((u + u) * (t * n)))
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    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_))))) <= 0.0) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	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_))))) <= 0.0:
		tmp = math.sqrt(((U + U) * (t * n)))
	else:
		tmp = math.sqrt(math.fabs((t * (U * (n + n)))))
	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_))))) <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	else
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	end
	return tmp
end

MATLAB

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_))))) <= 0.0)
		tmp = sqrt(((U + U) * (t * n)));
	else
		tmp = sqrt(abs((t * (U * (n + n)))));
	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], 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 35.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.8

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

   6. Applied rewrites 35.8%

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

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

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.8%

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

Alternative 10: 35.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.26 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.26e-101) (sqrt (* (* U (+ n n)) t)) (sqrt (* (+ U U) (* t n)))))

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 <= 1.26e-101) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = sqrt(((U + U) * (t * n)));
	}
	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 <= 1.26d-101) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = sqrt(((u + u) * (t * n)))
    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 <= 1.26e-101) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.sqrt(((U + U) * (t * n)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.26e-101:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.sqrt(((U + U) * (t * n)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.26e-101)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	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 <= 1.26e-101)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = sqrt(((U + U) * (t * n)));
	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, 1.26e-101], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.26e-101

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 35.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 35.7

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lift-+.f64 35.7

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

   6. Applied rewrites 35.7%

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

   if 1.26e-101 < l

   1. Initial program 49.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 35.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.8

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

   6. Applied rewrites 35.8%

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

Alternative 11: 35.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

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(((u * (n + n)) * t))
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(((U * (n + n)) * t));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.6%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 35.8

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

4. Applied rewrites 35.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 35.7

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 35.7

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lift-+.f64 35.7

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

6. Applied rewrites 35.7%

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

Reproduce

herbie shell --seed 2025156 
(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*))))))