Toniolo and Linder, Equation (13)

Percentage Accurate: 51.0% → 59.8%

Time: 9.0s

Alternatives: 14

Speedup: 2.4×

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: 51.0% 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: 59.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ \mathbf{if}\;n \leq -2.45 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t\_1 - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t\_1 - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t)))
   (if (<= n -2.45e-56)
     (sqrt
      (* (* n 2.0) (* U (- t_1 (* (- U U*) (* (* (/ l Om) (/ l Om)) n))))))
     (if (<= n 4.2e-268)
       (sqrt (* (* (* t_1 n) U) 2.0))
       (*
        (sqrt (* n 2.0))
        (sqrt (* U (- t_1 (* (- U U*) (* (pow (/ l Om) 2.0) n))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(-2.0, (l * (l / Om)), t);
	double tmp;
	if (n <= -2.45e-56) {
		tmp = sqrt(((n * 2.0) * (U * (t_1 - ((U - U_42_) * (((l / Om) * (l / Om)) * n))))));
	} else if (n <= 4.2e-268) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t_1 - ((U - U_42_) * (pow((l / Om), 2.0) * n)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	tmp = 0.0
	if (n <= -2.45e-56)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t_1 - Float64(Float64(U - U_42_) * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * n))))));
	elseif (n <= 4.2e-268)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t_1 - Float64(Float64(U - U_42_) * Float64((Float64(l / Om) ^ 2.0) * n))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[n, -2.45e-56], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t$95$1 - N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.2e-268], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t$95$1 - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.45e-56

   1. Initial program 55.9%

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

   3. Applied rewrites 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if -2.45e-56 < n < 4.19999999999999996e-268

   1. Initial program 44.8%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 52.8

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

   4. Applied rewrites 52.8%

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

   if 4.19999999999999996e-268 < n

   1. Initial program 51.7%

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

   3. Applied rewrites 55.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

   5. Applied rewrites 64.3%

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

Alternative 2: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{Om} \cdot \frac{\ell}{Om}\\ t_2 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t\_3 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_5 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot U\right)}{Om} \cdot \frac{\left(U - U*\right) \cdot n}{Om}, -2, \left(t\_5 \cdot U\right) \cdot 2\right) \cdot n}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(t\_2 - \left(n \cdot t\_1\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t\_5 - \left(U - U*\right) \cdot \left(t\_1 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\ell \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot n\right)\right)}{Om \cdot Om}\right)\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l Om) (/ l Om)))
        (t_2 (- t (* 2.0 (/ (* l l) Om))))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (- t_2 (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        (t_5 (fma -2.0 (* l (/ l Om)) t)))
   (if (<= t_4 0.0)
     (sqrt
      (*
       (fma
        (* (/ (* l (* l U)) Om) (/ (* (- U U*) n) Om))
        -2.0
        (* (* t_5 U) 2.0))
       n))
     (if (<= t_4 4e+295)
       (sqrt (* t_3 (- t_2 (* (* n t_1) (- U U*)))))
       (if (<= t_4 INFINITY)
         (sqrt (* (* n 2.0) (* U (- t_5 (* (- U U*) (* t_1 n))))))
         (sqrt (* (* l (* l (/ (* 2.0 (* U (* U* n))) (* Om Om)))) n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double t_2 = t - (2.0 * ((l * l) / Om));
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * (t_2 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double t_5 = fma(-2.0, (l * (l / Om)), t);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((fma((((l * (l * U)) / Om) * (((U - U_42_) * n) / Om)), -2.0, ((t_5 * U) * 2.0)) * n));
	} else if (t_4 <= 4e+295) {
		tmp = sqrt((t_3 * (t_2 - ((n * t_1) * (U - U_42_)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((n * 2.0) * (U * (t_5 - ((U - U_42_) * (t_1 * n))))));
	} else {
		tmp = sqrt(((l * (l * ((2.0 * (U * (U_42_ * n))) / (Om * Om)))) * n));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l / Om) * Float64(l / Om))
	t_2 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(t_2 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_5 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l * Float64(l * U)) / Om) * Float64(Float64(Float64(U - U_42_) * n) / Om)), -2.0, Float64(Float64(t_5 * U) * 2.0)) * n));
	elseif (t_4 <= 4e+295)
		tmp = sqrt(Float64(t_3 * Float64(t_2 - Float64(Float64(n * t_1) * Float64(U - U_42_)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t_5 - Float64(Float64(U - U_42_) * Float64(t_1 * n))))));
	else
		tmp = sqrt(Float64(Float64(l * Float64(l * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * n))) / Float64(Om * Om)))) * n));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(t$95$2 - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(N[(N[(N[(l * N[(l * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(t$95$5 * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 4e+295], N[Sqrt[N[(t$95$3 * N[(t$95$2 - N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t$95$5 - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(l * N[(N[(2.0 * N[(U * N[(U$42$ * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.4%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 35.2

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

   6. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   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*)))) < 3.9999999999999999e295

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 97.0

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

   3. Applied rewrites 97.0%

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

   if 3.9999999999999999e295 < (*.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 35.3%

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

   3. Applied rewrites 45.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 45.1

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

   5. Applied rewrites 45.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 31.1

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

   10. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 33.0

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

   12. Applied rewrites 33.0%

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

Alternative 3: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -1 \cdot \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(\ell \cdot \ell, n \cdot \frac{U - U*}{Om}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{Om}, t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\ell \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot n\right)\right)}{Om \cdot Om}\right)\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt
      (*
       (* n 2.0)
       (*
        U
        (+
         t
         (*
          -1.0
          (/ (fma 2.0 (* l l) (/ (* (* l l) (* n (- U U*))) Om)) Om))))))
     (if (<= t_2 4e+294)
       (sqrt
        (*
         t_1
         (fma
          -1.0
          (/ (fma (* l l) (* n (/ (- U U*) Om)) (* 2.0 (* l l))) Om)
          t)))
       (if (<= t_2 INFINITY)
         (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
         (sqrt (* (* l (* l (/ (* 2.0 (* U (* U* n))) (* Om Om)))) n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * 2.0) * (U * (t + (-1.0 * (fma(2.0, (l * l), (((l * l) * (n * (U - U_42_))) / Om)) / Om))))));
	} else if (t_2 <= 4e+294) {
		tmp = sqrt((t_1 * fma(-1.0, (fma((l * l), (n * ((U - U_42_) / Om)), (2.0 * (l * l))) / Om), t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((l * (l * ((2.0 * (U * (U_42_ * n))) / (Om * Om)))) * n));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(-1.0 * Float64(fma(2.0, Float64(l * l), Float64(Float64(Float64(l * l) * Float64(n * Float64(U - U_42_))) / Om)) / Om))))));
	elseif (t_2 <= 4e+294)
		tmp = sqrt(Float64(t_1 * fma(-1.0, Float64(fma(Float64(l * l), Float64(n * Float64(Float64(U - U_42_) / Om)), Float64(2.0 * Float64(l * l))) / Om), t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(l * Float64(l * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * n))) / Float64(Om * Om)))) * n));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(-1.0 * N[(N[(2.0 * N[(l * l), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+294], N[Sqrt[N[(t$95$1 * N[(-1.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(l * N[(N[(2.0 * N[(U * N[(U$42$ * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.4%

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

   3. Applied rewrites 36.2%

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

   4. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 35.4

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

   6. Applied rewrites 35.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000027e294

   1. Initial program 97.0%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 86.2%

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

   if 4.00000000000000027e294 < (*.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 35.4%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 38.9

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

   4. Applied rewrites 38.9%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 31.1

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

   10. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 33.0

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

   12. Applied rewrites 33.0%

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

Alternative 4: 55.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(\ell \cdot \ell, n \cdot \frac{U - U*}{Om}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{Om}, t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\ell \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot n\right)\right)}{Om \cdot Om}\right)\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt (* (* n 2.0) (* U t)))
     (if (<= t_2 4e+294)
       (sqrt
        (*
         t_1
         (fma
          -1.0
          (/ (fma (* l l) (* n (/ (- U U*) Om)) (* 2.0 (* l l))) Om)
          t)))
       (if (<= t_2 INFINITY)
         (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
         (sqrt (* (* l (* l (/ (* 2.0 (* U (* U* n))) (* Om Om)))) n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * 2.0) * (U * t)));
	} else if (t_2 <= 4e+294) {
		tmp = sqrt((t_1 * fma(-1.0, (fma((l * l), (n * ((U - U_42_) / Om)), (2.0 * (l * l))) / Om), t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((l * (l * ((2.0 * (U * (U_42_ * n))) / (Om * Om)))) * n));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * t)));
	elseif (t_2 <= 4e+294)
		tmp = sqrt(Float64(t_1 * fma(-1.0, Float64(fma(Float64(l * l), Float64(n * Float64(Float64(U - U_42_) / Om)), Float64(2.0 * Float64(l * l))) / Om), t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(l * Float64(l * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * n))) / Float64(Om * Om)))) * n));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+294], N[Sqrt[N[(t$95$1 * N[(-1.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(l * N[(N[(2.0 * N[(U * N[(U$42$ * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.4%

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

   3. Applied rewrites 36.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 30.2%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000027e294

   1. Initial program 97.0%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 86.2%

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

   if 4.00000000000000027e294 < (*.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 35.4%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 38.9

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

   4. Applied rewrites 38.9%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 31.1

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

   10. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 33.0

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

   12. Applied rewrites 33.0%

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

Alternative 5: 54.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, 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 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(l * N[(N[(2.0 * N[(U * N[(U$42$ * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $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 11.4%

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

   3. Applied rewrites 36.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 30.2%

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

   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 69.2%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.9

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

   4. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

   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 0.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 31.1

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

   10. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 33.0

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

   12. Applied rewrites 33.0%

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

Alternative 6: 44.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (* n 2.0) (* U t)))
     (if (<= t_1 1e+304)
       (sqrt (* (* (+ n n) U) t))
       (sqrt (* (* -4.0 (* U (* l (/ l Om)))) n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((n * 2.0) * (U * t)));
	} else if (t_1 <= 1e+304) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = sqrt(((-4.0 * (U * (l * (l / Om)))) * n));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((n * 2.0d0) * (u * t)))
    else if (t_1 <= 1d+304) then
        tmp = sqrt((((n + n) * u) * t))
    else
        tmp = sqrt((((-4.0d0) * (u * (l * (l / om)))) * n))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((n * 2.0) * (U * t)));
	} else if (t_1 <= 1e+304) {
		tmp = Math.sqrt((((n + n) * U) * t));
	} else {
		tmp = Math.sqrt(((-4.0 * (U * (l * (l / Om)))) * n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((n * 2.0) * (U * t)))
	elif t_1 <= 1e+304:
		tmp = math.sqrt((((n + n) * U) * t))
	else:
		tmp = math.sqrt(((-4.0 * (U * (l * (l / Om)))) * n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * t)));
	elseif (t_1 <= 1e+304)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(U * Float64(l * Float64(l / Om)))) * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((n * 2.0) * (U * t)));
	elseif (t_1 <= 1e+304)
		tmp = sqrt((((n + n) * U) * t));
	else
		tmp = sqrt(((-4.0 * (U * (l * (l / Om)))) * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, 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 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(U * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $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 11.4%

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

   3. Applied rewrites 36.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 30.2%

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

   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*)))) < 9.9999999999999994e303

   1. Initial program 97.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.7

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

   6. Applied rewrites 75.7%

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

   if 9.9999999999999994e303 < (*.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 23.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 29.1

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

   7. Applied rewrites 29.1%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 29.6

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

   10. Applied rewrites 29.6%

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

   11. Taylor expanded in n around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r/ N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 22.9

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

   13. Applied rewrites 22.9%

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

Alternative 7: 58.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], Infinity], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(l * N[(N[(2.0 * N[(U * N[(U$42$ * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $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*)))) < +inf.0

   1. Initial program 59.5%

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

   3. Applied rewrites 62.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 31.1

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

   10. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 33.0

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

   12. Applied rewrites 33.0%

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

Alternative 8: 50.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (* (sqrt (* n 2.0)) (sqrt (* U t)))
   (sqrt (* (* (+ n n) U) (fma -2.0 (* l (/ l Om)) t)))))

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $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 12.8%

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

   3. Applied rewrites 37.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 31.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 30.3

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

   8. Applied rewrites 30.3%

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

   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 56.6%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 53.3

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

   6. Applied rewrites 53.3%

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

Alternative 9: 38.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (* n 2.0) (* U t)))
   (sqrt (* (* (+ n n) U) t))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt(((n * 2.0d0) * (u * t)))
    else
        tmp = sqrt((((n + n) * u) * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt(((n * 2.0) * (U * t)));
	} else {
		tmp = Math.sqrt((((n + n) * U) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt(((n * 2.0) * (U * t)))
	else:
		tmp = math.sqrt((((n + n) * U) * t))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $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 12.8%

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

   3. Applied rewrites 37.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 31.3%

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

   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 56.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 39.8

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

   6. Applied rewrites 39.8%

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

Alternative 10: 38.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (* (+ n n) U) t))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((((n + n) * u) * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((n + n) * U) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((((n + n) * U) * t))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $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 12.8%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

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

   1. Initial program 56.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 39.8

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

   6. Applied rewrites 39.8%

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

Alternative 11: 52.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+20}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e+20)
   (sqrt (* (* n 2.0) (* U (- t (* (- U U*) (* (* (/ l Om) (/ l Om)) n))))))
   (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e+20) {
		tmp = sqrt(((n * 2.0) * (U * (t - ((U - U_42_) * (((l / Om) * (l / Om)) * n))))));
	} else {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e+20)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(U - U_42_) * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * n))))));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e+20], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1e20

   1. Initial program 54.9%

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

   3. Applied rewrites 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.0%

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

   if 1e20 < l

   1. Initial program 36.6%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 43.2

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

   4. Applied rewrites 43.2%

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

Alternative 12: 39.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot 2} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -4.2e-308)
   (sqrt (* (* n 2.0) (* U t)))
   (* (sqrt (* (* U n) 2.0)) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -4.2e-308) {
		tmp = sqrt(((n * 2.0) * (U * t)));
	} else {
		tmp = sqrt(((U * n) * 2.0)) * sqrt(t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-4.2d-308)) then
        tmp = sqrt(((n * 2.0d0) * (u * t)))
    else
        tmp = sqrt(((u * n) * 2.0d0)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -4.2e-308) {
		tmp = Math.sqrt(((n * 2.0) * (U * t)));
	} else {
		tmp = Math.sqrt(((U * n) * 2.0)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -4.2e-308:
		tmp = math.sqrt(((n * 2.0) * (U * t)))
	else:
		tmp = math.sqrt(((U * n) * 2.0)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -4.2e-308)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * t)));
	else
		tmp = Float64(sqrt(Float64(Float64(U * n) * 2.0)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -4.2e-308)
		tmp = sqrt(((n * 2.0) * (U * t)));
	else
		tmp = sqrt(((U * n) * 2.0)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -4.2e-308], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2e-308

   1. Initial program 51.4%

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

   3. Applied rewrites 55.4%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 36.4%

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

   if -4.2e-308 < t

   1. Initial program 50.6%

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

   3. Applied rewrites 52.7%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 43.1%

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

Alternative 13: 38.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.7 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 1.7e-268)
   (sqrt (* (* (+ n n) U) t))
   (* (sqrt (* n 2.0)) (sqrt (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1.7e-268) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 1.7d-268) then
        tmp = sqrt((((n + n) * u) * t))
    else
        tmp = sqrt((n * 2.0d0)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1.7e-268) {
		tmp = Math.sqrt((((n + n) * U) * t));
	} else {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 1.7e-268:
		tmp = math.sqrt((((n + n) * U) * t))
	else:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((U * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 1.7e-268)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 1.7e-268)
		tmp = sqrt((((n + n) * U) * t));
	else
		tmp = sqrt((n * 2.0)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 1.7e-268], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.7e-268

   1. Initial program 50.4%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 36.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 36.9

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

   6. Applied rewrites 36.9%

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

   if 1.7e-268 < n

   1. Initial program 51.6%

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

   3. Applied rewrites 55.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 35.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 40.6

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

   8. Applied rewrites 40.6%

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

Alternative 14: 36.4% accurate, 7.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* (+ n n) U) t)))

C

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

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((((n + n) * u) * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.0%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 36.4%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 36.4

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

6. Applied rewrites 36.4%

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

Reproduce

herbie shell --seed 2025099 
(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*))))))