Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 64.3%

Time: 7.9s

Alternatives: 19

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - t\_1\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (- (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 0.0)
     (pow (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0) 0.5)
     (if (<= t_3 5e+286)
       (sqrt (* t_2 (- (fma (* -2.0 l_m) (/ l_m Om) t) t_1)))
       (*
        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 t_1 = (n * pow((l_m / Om), 2.0)) * (U - U_42_);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) - t_1);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = pow((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0), 0.5);
	} else if (t_3 <= 5e+286) {
		tmp = sqrt((t_2 * (fma((-2.0 * l_m), (l_m / Om), t) - t_1)));
	} 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_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - t_1))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0) ^ 0.5;
	elseif (t_3 <= 5e+286)
		tmp = sqrt(Float64(t_2 * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - t_1)));
	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$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Power[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$3, 5e+286], N[Sqrt[N[(t$95$2 * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - t$95$1), $MachinePrecision]), $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 (*.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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000004e286

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

   if 5.0000000000000004e286 < (*.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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 28.1

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

   4. Applied rewrites 28.1%

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

Alternative 2: 64.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{t\_1}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (pow (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0) 0.5)
     (if (<= t_1 5e+286)
       (sqrt t_1)
       (*
        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 t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = pow((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0), 0.5);
	} else if (t_1 <= 5e+286) {
		tmp = sqrt(t_1);
	} 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_)
	t_1 = 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_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0) ^ 0.5;
	elseif (t_1 <= 5e+286)
		tmp = sqrt(t_1);
	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$42$_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 0.0], N[Power[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$1, 5e+286], N[Sqrt[t$95$1], $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 (*.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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000004e286

   1. Initial program 50.5%

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

   if 5.0000000000000004e286 < (*.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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 28.1

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

   4. Applied rewrites 28.1%

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

Alternative 3: 64.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\\ t_3 := t\_1 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t\_2 - \left(\frac{l\_m}{Om} \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (- t (* 2.0 (/ (* l_m l_m) Om))))
        (t_3 (* t_1 (- t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (pow (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0) 0.5)
     (if (<= t_3 5e+286)
       (sqrt (* t_1 (- t_2 (* (* (/ l_m Om) (* (/ l_m Om) n)) (- U 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 t_1 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l_m * l_m) / Om));
	double t_3 = t_1 * (t_2 - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = pow((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0), 0.5);
	} else if (t_3 <= 5e+286) {
		tmp = sqrt((t_1 * (t_2 - (((l_m / Om) * ((l_m / Om) * n)) * (U - U_42_)))));
	} 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_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om)))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0) ^ 0.5;
	elseif (t_3 <= 5e+286)
		tmp = sqrt(Float64(t_1 * Float64(t_2 - Float64(Float64(Float64(l_m / Om) * Float64(Float64(l_m / Om) * n)) * Float64(U - U_42_)))));
	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$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - 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$3, 0.0], N[Power[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$3, 5e+286], N[Sqrt[N[(t$95$1 * N[(t$95$2 - 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[(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 (*.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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000004e286

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 51.7

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

   3. Applied rewrites 51.7%

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

   if 5.0000000000000004e286 < (*.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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 28.1

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

   4. Applied rewrites 28.1%

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

Alternative 4: 63.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_2\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{l\_m \cdot l\_m}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, t\_2, t\right)\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(t\_1 - \left(\left(U - U*\right) \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* l_m (/ l_m Om)) -2.0 t))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 t_2)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (*
      (sqrt (+ n n))
      (sqrt
       (* (fma (- U* U) (* (/ (* l_m l_m) (* Om Om)) n) (fma -2.0 t_2 t)) U)))
     (if (<= t_3 INFINITY)
       (sqrt
        (* (- t_1 (* (* (- U U*) (* (/ l_m Om) n)) (/ l_m Om))) (* U (+ n n))))
       (pow (* (* (* t_1 n) U) 2.0) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((l_m * (l_m / Om)), -2.0, t);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_2)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma((U_42_ - U), (((l_m * l_m) / (Om * Om)) * n), fma(-2.0, t_2, t)) * U));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t_1 - (((U - U_42_) * ((l_m / Om) * n)) * (l_m / Om))) * (U * (n + n))));
	} else {
		tmp = pow((((t_1 * n) * U) * 2.0), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_2)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(U_42_ - U), Float64(Float64(Float64(l_m * l_m) / Float64(Om * Om)) * n), fma(-2.0, t_2, t)) * U)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(t_1 - Float64(Float64(Float64(U - U_42_) * Float64(Float64(l_m / Om) * n)) * Float64(l_m / Om))) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(Float64(Float64(t_1 * n) * U) * 2.0) ^ 0.5;
	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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$2), $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$3, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$2 + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(t$95$1 - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 26.9%

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

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

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(t\_2 - \left(\left(U - U*\right) \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(t\_2 \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (fma (* l_m (/ l_m Om)) -2.0 t)))
   (if (<= t_1 0.0)
     (* n (sqrt (* 2.0 (/ (* U t) n))))
     (if (<= t_1 INFINITY)
       (sqrt
        (* (- t_2 (* (* (- U U*) (* (/ l_m Om) n)) (/ l_m Om))) (* U (+ n n))))
       (pow (* (* (* t_2 n) U) 2.0) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = fma((l_m * (l_m / Om)), -2.0, t);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * sqrt((2.0 * ((U * t) / n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((t_2 - (((U - U_42_) * ((l_m / Om) * n)) * (l_m / Om))) * (U * (n + n))));
	} else {
		tmp = pow((((t_2 * n) * U) * 2.0), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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_)))))
	t_2 = fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(t_2 - Float64(Float64(Float64(U - U_42_) * Float64(Float64(l_m / Om) * n)) * Float64(l_m / Om))) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(Float64(Float64(t_2 * n) * U) * 2.0) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(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]}, Block[{t$95$2 = N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(t$95$2 - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(t$95$2 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

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

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-162}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t - \left(\left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \end{array} \end{array} \]

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-162)
     (* n (sqrt (* 2.0 (/ (* U t) n))))
     (if (<= t_2 2e+139)
       (sqrt
        (*
         t_1
         (fma
          (/ l_m Om)
          (* l_m -2.0)
          (- t (* (* (* l_m (/ l_m (* Om Om))) n) (- U U*))))))
       (pow (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0) 0.5)))))

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-162) {
		tmp = n * sqrt((2.0 * ((U * t) / n)));
	} else if (t_2 <= 2e+139) {
		tmp = sqrt((t_1 * fma((l_m / Om), (l_m * -2.0), (t - (((l_m * (l_m / (Om * Om))) * n) * (U - U_42_))))));
	} else {
		tmp = pow((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0), 0.5);
	}
	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-162)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	elseif (t_2 <= 2e+139)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), Float64(l_m * -2.0), Float64(t - Float64(Float64(Float64(l_m * Float64(l_m / Float64(Om * Om))) * n) * Float64(U - U_42_))))));
	else
		tmp = Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0) ^ 0.5;
	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-162], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+139], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + N[(t - N[(N[(N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $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*))))) < 5.00000000000000014e-162

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

   if 5.00000000000000014e-162 < (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*))))) < 2.00000000000000007e139

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 51.3%

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

   if 2.00000000000000007e139 < (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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-162}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right), n, t\_2\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(t\_2 \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (fma (* l_m (/ l_m Om)) -2.0 t)))
   (if (<= t_1 5e-162)
     (* n (sqrt (* 2.0 (/ (* U t) n))))
     (if (<= t_1 2e+139)
       (sqrt
        (* (fma (* (- U* U) (* l_m (/ l_m (* Om Om)))) n t_2) (* U (+ n n))))
       (pow (* (* (* t_2 n) U) 2.0) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = fma((l_m * (l_m / Om)), -2.0, t);
	double tmp;
	if (t_1 <= 5e-162) {
		tmp = n * sqrt((2.0 * ((U * t) / n)));
	} else if (t_1 <= 2e+139) {
		tmp = sqrt((fma(((U_42_ - U) * (l_m * (l_m / (Om * Om)))), n, t_2) * (U * (n + n))));
	} else {
		tmp = pow((((t_2 * n) * U) * 2.0), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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_)))))
	t_2 = fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)
	tmp = 0.0
	if (t_1 <= 5e-162)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	elseif (t_1 <= 2e+139)
		tmp = sqrt(Float64(fma(Float64(Float64(U_42_ - U) * Float64(l_m * Float64(l_m / Float64(Om * Om)))), n, t_2) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(Float64(Float64(t_2 * n) * U) * 2.0) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(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]}, Block[{t$95$2 = N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-162], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+139], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n + t$95$2), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(t$95$2 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $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*))))) < 5.00000000000000014e-162

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

   if 5.00000000000000014e-162 < (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*))))) < 2.00000000000000007e139

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.5%

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

   5. Applied rewrites 49.1%

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

   if 2.00000000000000007e139 < (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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \left(U \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* n (sqrt (* 2.0 (/ (* U t) n))))
     (if (<= t_1 2e+139)
       (sqrt
        (*
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (* U (* n (/ l_m Om))) (/ l_m Om)))
         (* (+ n n) U)))
       (pow (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * sqrt((2.0 * ((U * t) / n)));
	} else if (t_1 <= 2e+139) {
		tmp = sqrt(((fma((-2.0 * l_m), (l_m / Om), t) - ((U * (n * (l_m / Om))) * (l_m / Om))) * ((n + n) * U)));
	} else {
		tmp = pow((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	elseif (t_1 <= 2e+139)
		tmp = sqrt(Float64(Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(U * Float64(n * Float64(l_m / Om))) * Float64(l_m / Om))) * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(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]}, If[LessEqual[t$95$1, 0.0], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+139], N[Sqrt[N[(N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(U * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000007e139

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.2

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

   3. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in U around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.7

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

   8. Applied rewrites 39.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.7

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

   10. Applied rewrites 41.4%

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

   if 2.00000000000000007e139 < (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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* n (sqrt (* 2.0 (/ (* U t) n))))
     (if (<= t_1 2e+139)
       (sqrt (* (* (+ n n) U) (fma (* -2.0 l_m) (/ l_m Om) t)))
       (pow (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * sqrt((2.0 * ((U * t) / n)));
	} else if (t_1 <= 2e+139) {
		tmp = sqrt((((n + n) * U) * fma((-2.0 * l_m), (l_m / Om), t)));
	} else {
		tmp = pow((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	elseif (t_1 <= 2e+139)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(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]}, If[LessEqual[t$95$1, 0.0], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+139], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000007e139

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 46.9

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

   6. Applied rewrites 46.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

   8. Applied rewrites 47.0%

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

   if 2.00000000000000007e139 < (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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 50.2

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

   6. Applied rewrites 53.9%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U}{Om} \cdot -4\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0))
     (if (<= t_1 INFINITY)
       (sqrt (* (* (+ n n) U) (fma (* -2.0 l_m) (/ l_m Om) t)))
       (pow (* (/ (* (* (* l_m l_m) n) U) Om) -4.0) 0.5)))))

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) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((n + n) * U) * fma((-2.0 * l_m), (l_m / Om), t)));
	} else {
		tmp = pow((((((l_m * l_m) * n) * U) / Om) * -4.0), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * U) / Om) * -4.0) ^ 0.5;
	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[(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]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.2

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

   6. Applied rewrites 47.9%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 46.9

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

   6. Applied rewrites 46.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

   8. Applied rewrites 47.0%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 14.2

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

   7. Applied rewrites 14.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{\frac{1}{2}}} \]

      3. lower-pow.f64 19.6

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

   9. Applied rewrites 19.6%

      \[\leadsto \color{blue}{{\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot -4\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* n (sqrt (* 2.0 (/ (* U t) n))))
     (if (<= t_1 2e+139)
       (sqrt (* (* (+ n n) U) (fma (* -2.0 l_m) (/ l_m Om) t)))
       (sqrt (* (* (* (fma (* l_m (/ l_m Om)) -2.0 t) n) U) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * sqrt((2.0 * ((U * t) / n)));
	} else if (t_1 <= 2e+139) {
		tmp = sqrt((((n + n) * U) * fma((-2.0 * l_m), (l_m / Om), t)));
	} else {
		tmp = sqrt((((fma((l_m * (l_m / Om)), -2.0, t) * n) * U) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	elseif (t_1 <= 2e+139)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) * n) * U) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[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]}, If[LessEqual[t$95$1, 0.0], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+139], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000007e139

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 46.9

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

   6. Applied rewrites 46.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

   8. Applied rewrites 47.0%

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

   if 2.00000000000000007e139 < (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 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.2

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

   6. Applied rewrites 47.9%

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

Alternative 12: 49.7% accurate, 0.7× speedup

Math

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

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)
   (* n (sqrt (* 2.0 (/ (* U t) n))))
   (sqrt (* (* (+ n n) U) (fma (* -2.0 l_m) (/ l_m Om) t)))))

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 = n * sqrt((2.0 * ((U * t) / n)));
	} else {
		tmp = sqrt((((n + n) * U) * fma((-2.0 * l_m), (l_m / Om), t)));
	}
	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 = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	else
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $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 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 46.9

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

   6. Applied rewrites 46.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

   8. Applied rewrites 47.0%

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

Alternative 13: 39.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.45 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.45e+85)
   (sqrt (* (* U (+ n n)) t))
   (sqrt (/ (* -4.0 (* (* (* l_m l_m) n) U)) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.45e+85) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = sqrt(((-4.0 * (((l_m * l_m) * n) * U)) / Om));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.45d+85) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = sqrt((((-4.0d0) * (((l_m * l_m) * n) * u)) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.45e+85) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.sqrt(((-4.0 * (((l_m * l_m) * n) * U)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.45e+85:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.sqrt(((-4.0 * (((l_m * l_m) * n) * U)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.45e+85)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(Float64(l_m * l_m) * n) * U)) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.45e+85)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = sqrt(((-4.0 * (((l_m * l_m) * n) * U)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.45e+85], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.44999999999999999e85

   1. Initial program 50.5%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.6

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.8

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

   6. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 35.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 35.9

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 35.9

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

   8. Applied rewrites 35.9%

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

   if 1.44999999999999999e85 < l

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 14.2

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

   7. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 14.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 14.2

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 14.2

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

   9. Applied rewrites 14.2%

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

Alternative 14: 39.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.45 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{\left(U \cdot \left(l\_m \cdot l\_m\right)\right) \cdot n}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.45e+85)
   (sqrt (* (* U (+ n n)) t))
   (sqrt (* -4.0 (/ (* (* U (* l_m l_m)) n) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.45e+85) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = sqrt((-4.0 * (((U * (l_m * l_m)) * n) / Om)));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.45d+85) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = sqrt(((-4.0d0) * (((u * (l_m * l_m)) * n) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.45e+85) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.sqrt((-4.0 * (((U * (l_m * l_m)) * n) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.45e+85:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.sqrt((-4.0 * (((U * (l_m * l_m)) * n) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.45e+85)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(Float64(U * Float64(l_m * l_m)) * n) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.45e+85)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = sqrt((-4.0 * (((U * (l_m * l_m)) * n) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.45e+85], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.44999999999999999e85

   1. Initial program 50.5%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.6

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.8

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

   6. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 35.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 35.9

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 35.9

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

   8. Applied rewrites 35.9%

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

   if 1.44999999999999999e85 < l

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.2

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      5. lower-pow.f64 14.2

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

   7. Applied rewrites 14.2%

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot {\ell}^{2}\right) \cdot n}{Om}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot {\ell}^{2}\right) \cdot n}{Om}} \]

      5. lower-*.f64 13.8

         \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot {\ell}^{2}\right) \cdot n}{Om}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot {\ell}^{2}\right) \cdot n}{Om}} \]

      7. pow2 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}} \]

      8. lower-*.f64 13.8

         \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}} \]

   9. Applied rewrites 13.8%

      \[\leadsto \sqrt{-4 \cdot \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \end{array} \]

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)
   (* n (sqrt (* 2.0 (/ (* U t) n))))
   (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_) {
	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 = n * sqrt((2.0 * ((U * t) / n)));
	} else {
		tmp = sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: 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 = n * sqrt((2.0d0 * ((u * t) / n)))
    else
        tmp = sqrt(((u * (n + n)) * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double 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 = n * Math.sqrt((2.0 * ((U * t) / n)));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	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 = n * math.sqrt((2.0 * ((U * t) / n)))
	else:
		tmp = math.sqrt(((U * (n + n)) * t))
	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 = Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	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 = n * sqrt((2.0 * ((U * t) / n)));
	else
		tmp = sqrt(((U * (n + n)) * t));
	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[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + \color{blue}{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + \color{blue}{-2} \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{Om} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

   4. Applied rewrites 32.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      3. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

      5. lower-*.f64 18.2

         \[\leadsto n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}} \]

   7. Applied rewrites 18.2%

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot t}{n}}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.6

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.8

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lower-+.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 38.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-323}:\\ \;\;\;\;U \cdot \sqrt{2 \cdot \frac{n \cdot t}{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      2e-323)
   (* U (sqrt (* 2.0 (/ (* n t) U))))
   (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_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 2e-323) {
		tmp = U * sqrt((2.0 * ((n * t) / U)));
	} else {
		tmp = sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))) <= 2d-323) then
        tmp = u * sqrt((2.0d0 * ((n * t) / u)))
    else
        tmp = sqrt(((u * (n + n)) * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 2e-323) {
		tmp = U * Math.sqrt((2.0 * ((n * t) / U)));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 2e-323:
		tmp = U * math.sqrt((2.0 * ((n * t) / U)))
	else:
		tmp = math.sqrt(((U * (n + n)) * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (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_)))) <= 2e-323)
		tmp = Float64(U * sqrt(Float64(2.0 * Float64(Float64(n * t) / U))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))) <= 2e-323)
		tmp = U * sqrt((2.0 * ((n * t) / U)));
	else
		tmp = sqrt(((U * (n + n)) * t));
	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[(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], 2e-323], N[(U * N[Sqrt[N[(2.0 * N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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*)))) < 1.97626e-323

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + \color{blue}{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + \color{blue}{-2} \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{Om} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

   4. Applied rewrites 32.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

   5. Taylor expanded in U around inf

      \[\leadsto U \cdot \color{blue}{\sqrt{2 \cdot \frac{n \cdot t}{U}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto U \cdot \sqrt{2 \cdot \frac{n \cdot t}{U}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto U \cdot \sqrt{2 \cdot \frac{n \cdot t}{U}} \]

      3. lower-*.f64 N/A

         \[\leadsto U \cdot \sqrt{2 \cdot \frac{n \cdot t}{U}} \]

      4. lower-/.f64 N/A

         \[\leadsto U \cdot \sqrt{2 \cdot \frac{n \cdot t}{U}} \]

      5. lower-*.f64 19.4

         \[\leadsto U \cdot \sqrt{2 \cdot \frac{n \cdot t}{U}} \]

   7. Applied rewrites 19.4%

      \[\leadsto U \cdot \color{blue}{\sqrt{2 \cdot \frac{n \cdot t}{U}}} \]

   if 1.97626e-323 < (*.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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.6

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.8

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lower-+.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 38.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}\\ \end{array} \end{array} \]

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*)))))
      2e-157)
   (sqrt (* U (* (+ n n) t)))
   (sqrt (* 2.0 (* (* U n) t)))))

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_))))) <= 2e-157) {
		tmp = sqrt((U * ((n + n) * t)));
	} else {
		tmp = sqrt((2.0 * ((U * n) * t)));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42))))) <= 2d-157) then
        tmp = sqrt((u * ((n + n) * t)))
    else
        tmp = sqrt((2.0d0 * ((u * n) * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double 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_))))) <= 2e-157) {
		tmp = Math.sqrt((U * ((n + n) * t)));
	} else {
		tmp = Math.sqrt((2.0 * ((U * n) * t)));
	}
	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_))))) <= 2e-157:
		tmp = math.sqrt((U * ((n + n) * t)))
	else:
		tmp = math.sqrt((2.0 * ((U * n) * t)))
	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_))))) <= 2e-157)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * n) * t)));
	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_))))) <= 2e-157)
		tmp = sqrt((U * ((n + n) * t)));
	else
		tmp = sqrt((2.0 * ((U * n) * t)));
	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], 2e-157], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999989e-157

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.6

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.8

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lower-+.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 35.6

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 35.6

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 35.6%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   if 1.99999999999999989e-157 < (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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.6

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.8

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \end{array} \]

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 (* (+ n n) t)))
   (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_) {
	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 * ((n + n) * t)));
	} else {
		tmp = sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: 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 * ((n + n) * t)))
    else
        tmp = sqrt(((u * (n + n)) * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double 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 * ((n + n) * t)));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	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 * ((n + n) * t)))
	else:
		tmp = math.sqrt(((U * (n + n)) * t))
	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(U * Float64(Float64(n + n) * t)));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	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 * ((n + n) * t)));
	else
		tmp = sqrt(((U * (n + n)) * t));
	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[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.6

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.8

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lower-+.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 35.6

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 35.6

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 35.6%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 35.6

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.8

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lower-+.f64 35.9

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 35.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \end{array} \]

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(U * Float64(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[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

2. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lower-*.f64 35.6

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 35.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. lower-*.f64 35.8

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

6. Applied rewrites 35.8%

   \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   9. lower-*.f64 35.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   12. lower-*.f64 35.9

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   14. count-2-rev N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   15. lower-+.f64 35.9

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

8. Applied rewrites 35.9%

   \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   3. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

   8. lower-*.f64 35.6

      \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

   10. count-2-rev N/A

       \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   11. lift-+.f64 35.6

       \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

10. Applied rewrites 35.6%

    \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025151 
(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*))))))