Toniolo and Linder, Equation (13)

Percentage Accurate: 49.7% → 61.3%

Time: 9.7s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;n \leq -8.5 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \ell\right) \cdot \ell}{Om}, U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(n + n\right)}^{0.5} \cdot {\left(U \cdot t\_1\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (-
          (fma -2.0 (* l (/ l Om)) t)
          (* n (* (* (/ l Om) (/ l Om)) (- U U*))))))
   (if (<= n -8.5e+48)
     (sqrt (* (+ n n) (* U (- t (* n (* (/ l Om) (* (/ l Om) (- U U*))))))))
     (if (<= n -8e-179)
       (sqrt (* t_1 (* (+ n n) U)))
       (if (<= n 3.9e-268)
         (sqrt (* (+ n n) (fma -2.0 (/ (* (* U l) l) Om) (* U t))))
         (* (pow (+ n n) 0.5) (pow (* U t_1) 0.5)))))))

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) - (n * (((l / Om) * (l / Om)) * (U - U_42_)));
	double tmp;
	if (n <= -8.5e+48) {
		tmp = sqrt(((n + n) * (U * (t - (n * ((l / Om) * ((l / Om) * (U - U_42_))))))));
	} else if (n <= -8e-179) {
		tmp = sqrt((t_1 * ((n + n) * U)));
	} else if (n <= 3.9e-268) {
		tmp = sqrt(((n + n) * fma(-2.0, (((U * l) * l) / Om), (U * t))));
	} else {
		tmp = pow((n + n), 0.5) * pow((U * t_1), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_))))
	tmp = 0.0
	if (n <= -8.5e+48)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(n * Float64(Float64(l / Om) * Float64(Float64(l / Om) * Float64(U - U_42_))))))));
	elseif (n <= -8e-179)
		tmp = sqrt(Float64(t_1 * Float64(Float64(n + n) * U)));
	elseif (n <= 3.9e-268)
		tmp = sqrt(Float64(Float64(n + n) * fma(-2.0, Float64(Float64(Float64(U * l) * l) / Om), Float64(U * t))));
	else
		tmp = Float64((Float64(n + n) ^ 0.5) * (Float64(U * t_1) ^ 0.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8.5e+48], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(n * N[(N[(l / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -8e-179], N[Sqrt[N[(t$95$1 * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.9e-268], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(-2.0 * N[(N[(N[(U * l), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(n + n), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(U * t$95$1), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.5000000000000001e48

   1. Initial program 57.1%

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

   3. Applied rewrites 58.4%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if -8.5000000000000001e48 < n < -8.0000000000000002e-179

   1. Initial program 50.8%

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

   3. Applied rewrites 55.0%

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

   if -8.0000000000000002e-179 < n < 3.8999999999999998e-268

   1. Initial program 36.9%

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

   3. Applied rewrites 41.9%

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

   4. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 39.3

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

   6. Applied rewrites 39.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.3

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

   8. Applied rewrites 43.3%

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

   if 3.8999999999999998e-268 < n

   1. Initial program 50.9%

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

   3. Applied rewrites 63.0%

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

Alternative 2: 61.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(t\_1 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_4 := \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-116}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t\_1 - \left(n \cdot {\left({\left(\frac{\ell}{Om}\right)}^{-2}\right)}^{-1}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (* l l) Om))))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (- t_1 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_4
         (sqrt
          (*
           (+ n n)
           (*
            U
            (-
             (fma -2.0 (* l (/ l Om)) t)
             (* n (* (* (/ l Om) (/ l Om)) (- U U*)))))))))
   (if (<= t_3 2e-116)
     t_4
     (if (<= t_3 2e+151)
       (sqrt (* t_2 (- t_1 (* (* n (pow (pow (/ l Om) -2.0) -1.0)) (- U U*)))))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt
          (*
           (* (* (* (* l l) n) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
           -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (2.0 * ((l * l) / Om));
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * (t_1 - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_4 = sqrt(((n + n) * (U * (fma(-2.0, (l * (l / Om)), t) - (n * (((l / Om) * (l / Om)) * (U - U_42_)))))));
	double tmp;
	if (t_3 <= 2e-116) {
		tmp = t_4;
	} else if (t_3 <= 2e+151) {
		tmp = sqrt((t_2 * (t_1 - ((n * pow(pow((l / Om), -2.0), -1.0)) * (U - U_42_)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((((((l * l) * n) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(t_1 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_4 = sqrt(Float64(Float64(n + n) * Float64(U * Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))))))
	tmp = 0.0
	if (t_3 <= 2e-116)
		tmp = t_4;
	elseif (t_3 <= 2e+151)
		tmp = sqrt(Float64(t_2 * Float64(t_1 - Float64(Float64(n * ((Float64(l / Om) ^ -2.0) ^ -1.0)) * Float64(U - U_42_)))));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(t$95$1 - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-116], t$95$4, If[LessEqual[t$95$3, 2e+151], N[Sqrt[N[(t$95$2 * N[(t$95$1 - N[(N[(n * N[Power[N[Power[N[(l / Om), $MachinePrecision], -2.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2e-116 or 2.00000000000000003e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 30.1%

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

   3. Applied rewrites 43.7%

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

   if 2e-116 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 98.0%

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

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

      4. pow-unpow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 97.9

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

   3. Applied rewrites 97.9%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 31.0%

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

Alternative 3: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-116}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2
         (sqrt
          (*
           (+ n n)
           (*
            U
            (-
             (fma -2.0 (* l (/ l Om)) t)
             (* n (* (* (/ l Om) (/ l Om)) (- U U*)))))))))
   (if (<= t_1 2e-116)
     t_2
     (if (<= t_1 2e+151)
       t_1
       (if (<= t_1 INFINITY)
         t_2
         (sqrt
          (*
           (* (* (* (* l l) n) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
           -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt(((n + n) * (U * (fma(-2.0, (l * (l / Om)), t) - (n * (((l / Om) * (l / Om)) * (U - U_42_)))))));
	double tmp;
	if (t_1 <= 2e-116) {
		tmp = t_2;
	} else if (t_1 <= 2e+151) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = sqrt((((((l * l) * n) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	t_2 = sqrt(Float64(Float64(n + n) * Float64(U * Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))))))
	tmp = 0.0
	if (t_1 <= 2e-116)
		tmp = t_2;
	elseif (t_1 <= 2e+151)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, Block[{t$95$2 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-116], t$95$2, If[LessEqual[t$95$1, 2e+151], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2e-116 or 2.00000000000000003e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 30.1%

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

   3. Applied rewrites 43.7%

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

   if 2e-116 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 98.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 31.0%

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

Alternative 4: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_3 := \frac{\ell}{Om} \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t\_1 - n \cdot \left(t\_3 \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(t\_1 - n \cdot \left(t\_3 \cdot \left(-U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_3 (* (/ l Om) (/ l Om))))
   (if (<= t_2 2e-116)
     (sqrt (* (+ n n) (* U (- t_1 (* n (* t_3 (- U U*)))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (- t_1 (* n (* t_3 (- U*)))) (* (+ n n) U)))
       (sqrt
        (*
         (* (* (* (* l l) n) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
         -2.0))))))

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 t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_3 = (l / Om) * (l / Om);
	double tmp;
	if (t_2 <= 2e-116) {
		tmp = sqrt(((n + n) * (U * (t_1 - (n * (t_3 * (U - U_42_)))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((t_1 - (n * (t_3 * -U_42_))) * ((n + n) * U)));
	} else {
		tmp = sqrt((((((l * l) * n) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = 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_)))))
	t_3 = Float64(Float64(l / Om) * Float64(l / Om))
	tmp = 0.0
	if (t_2 <= 2e-116)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t_1 - Float64(n * Float64(t_3 * Float64(U - U_42_)))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(t_1 - Float64(n * Float64(t_3 * Float64(-U_42_)))) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	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]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-116], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t$95$1 - N[(n * N[(t$95$3 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(t$95$1 - N[(n * N[(t$95$3 * (-U$42$)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2e-116

   1. Initial program 26.5%

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

   3. Applied rewrites 45.7%

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

   if 2e-116 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 67.9%

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

   3. Applied rewrites 71.2%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.6

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

   6. Applied rewrites 71.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 31.0%

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

Alternative 5: 58.3% accurate, 0.3× speedup

Math

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

FPCore

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

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   3. Applied rewrites 37.7%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 34.0%

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

   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*))))) < +inf.0

   1. Initial program 69.0%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.5

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

   6. Applied rewrites 72.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 31.0%

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

Alternative 6: 56.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -8 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;U* \leq 1.15 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -8e+88)
   (sqrt (* (* (* 2.0 n) U) (- t (* (* n (* (/ l Om) (/ l Om))) (- U U*)))))
   (if (<= U* 1.15e+50)
     (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
     (sqrt (* (+ n n) (* U (- t (* n (* (/ l Om) (* (/ l Om) (- U U*)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -8e+88) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))));
	} else if (U_42_ <= 1.15e+50) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((n + n) * (U * (t - (n * ((l / Om) * ((l / Om) * (U - U_42_))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -8e+88)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(U - U_42_)))));
	elseif (U_42_ <= 1.15e+50)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(n * Float64(Float64(l / Om) * Float64(Float64(l / Om) * Float64(U - U_42_))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -8e+88], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 1.15e+50], 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[(n + n), $MachinePrecision] * N[(U * N[(t - N[(n * N[(N[(l / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U* < -7.99999999999999968e88

   1. Initial program 47.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 55.0%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 55.0

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

   6. Applied rewrites 55.0%

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

   if -7.99999999999999968e88 < U* < 1.14999999999999998e50

   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 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 53.6

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

   4. Applied rewrites 53.6%

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

   if 1.14999999999999998e50 < U*

   1. Initial program 49.8%

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

   3. Applied rewrites 51.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 56.4

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

   8. Applied rewrites 56.4%

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

Alternative 7: 56.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -8 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(t - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{elif}\;U* \leq 1.15 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -8e+88)
   (sqrt (* (- t (* n (* (* (/ l Om) (/ l Om)) (- U U*)))) (* (+ n n) U)))
   (if (<= U* 1.15e+50)
     (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
     (sqrt (* (+ n n) (* U (- t (* n (* (/ l Om) (* (/ l Om) (- U U*)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -8e+88) {
		tmp = sqrt(((t - (n * (((l / Om) * (l / Om)) * (U - U_42_)))) * ((n + n) * U)));
	} else if (U_42_ <= 1.15e+50) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((n + n) * (U * (t - (n * ((l / Om) * ((l / Om) * (U - U_42_))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -8e+88)
		tmp = sqrt(Float64(Float64(t - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))) * Float64(Float64(n + n) * U)));
	elseif (U_42_ <= 1.15e+50)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(n * Float64(Float64(l / Om) * Float64(Float64(l / Om) * Float64(U - U_42_))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -8e+88], N[Sqrt[N[(N[(t - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 1.15e+50], 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[(n + n), $MachinePrecision] * N[(U * N[(t - N[(n * N[(N[(l / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U* < -7.99999999999999968e88

   1. Initial program 47.6%

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

   3. Applied rewrites 49.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 53.6%

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

   if -7.99999999999999968e88 < U* < 1.14999999999999998e50

   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 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 53.6

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

   4. Applied rewrites 53.6%

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

   if 1.14999999999999998e50 < U*

   1. Initial program 49.8%

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

   3. Applied rewrites 51.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 56.4

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

   8. Applied rewrites 56.4%

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

Alternative 8: 54.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -45:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (+ n n) (* U (- t (* n (* (/ l Om) (* (/ l Om) (- U U*))))))))))
   (if (<= n -45.0)
     t_1
     (if (<= n 1.25e-113)
       (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n + n) * (U * (t - (n * ((l / Om) * ((l / Om) * (U - U_42_))))))));
	double tmp;
	if (n <= -45.0) {
		tmp = t_1;
	} else if (n <= 1.25e-113) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(n * Float64(Float64(l / Om) * Float64(Float64(l / Om) * Float64(U - U_42_))))))))
	tmp = 0.0
	if (n <= -45.0)
		tmp = t_1;
	elseif (n <= 1.25e-113)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(n * N[(N[(l / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -45.0], t$95$1, If[LessEqual[n, 1.25e-113], 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], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -45 or 1.2499999999999999e-113 < n

   1. Initial program 55.8%

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

   3. Applied rewrites 58.4%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -45 < n < 1.2499999999999999e-113

   1. Initial program 42.6%

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

   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 52.3

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

   4. Applied rewrites 52.3%

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

Alternative 9: 54.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -45:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (+ n n) (* U (- t (* n (* (* (/ l Om) (/ l Om)) (- U U*)))))))))
   (if (<= n -45.0)
     t_1
     (if (<= n 1.25e-113)
       (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n + n) * (U * (t - (n * (((l / Om) * (l / Om)) * (U - U_42_)))))));
	double tmp;
	if (n <= -45.0) {
		tmp = t_1;
	} else if (n <= 1.25e-113) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))))))
	tmp = 0.0
	if (n <= -45.0)
		tmp = t_1;
	elseif (n <= 1.25e-113)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -45.0], t$95$1, If[LessEqual[n, 1.25e-113], 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], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -45 or 1.2499999999999999e-113 < n

   1. Initial program 55.8%

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

   3. Applied rewrites 58.4%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 59.4%

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

   if -45 < n < 1.2499999999999999e-113

   1. Initial program 42.6%

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

   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 52.3

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

   4. Applied rewrites 52.3%

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

Alternative 10: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U* \cdot U}{Om} \cdot \frac{\left(n \cdot n\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* t (+ n n))) (sqrt U))
     (if (<= t_1 5e+153)
       (sqrt (* (fma (* (/ l Om) l) -2.0 t) (* (+ n n) U)))
       (if (<= t_1 INFINITY)
         (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
         (sqrt (* (* (/ (* U* U) Om) (/ (* (* n n) (* l l)) Om)) 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((fma(((l / Om) * l), -2.0, t) * ((n + n) * U)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((((U_42_ * U) / Om) * (((n * n) * (l * l)) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(Float64(n + n) * U)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U_42_ * U) / Om) * Float64(Float64(Float64(n * n) * Float64(l * l)) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 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[(N[(N[(U$42$ * U), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(n * n), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 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 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 31.6%

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

   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 35.9

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

   4. Applied rewrites 35.9%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.0

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

   6. Applied rewrites 5.0%

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

   7. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 22.5

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

   9. Applied rewrites 22.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-/.f64 26.0

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

   11. Applied rewrites 26.0%

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

Alternative 11: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om} \cdot \left(n + n\right)\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* t (+ n n))) (sqrt U))
     (if (<= t_1 5e+153)
       (sqrt (* (fma (* (/ l Om) l) -2.0 t) (* (+ n n) U)))
       (if (<= t_1 INFINITY)
         (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
         (sqrt (* (* (/ (* U* (* (* l l) n)) (* Om Om)) (+ n n)) U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((fma(((l / Om) * l), -2.0, t) * ((n + n) * U)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((((U_42_ * ((l * l) * n)) / (Om * Om)) * (n + n)) * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(Float64(n + n) * U)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * Om)) * Float64(n + n)) * U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 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[(N[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 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 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 31.6%

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

   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 35.9

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

   4. Applied rewrites 35.9%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 9.1

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

   8. Applied rewrites 9.1%

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

   9. Taylor expanded in U* around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 26.4

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

   11. Applied rewrites 26.4%

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

Alternative 12: 53.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \left(n + n\right) \cdot U\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot t\_2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om} \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2 (* (+ n n) U)))
   (if (<= t_1 0.0)
     (* (sqrt (* t (+ n n))) (sqrt U))
     (if (<= t_1 5e+153)
       (sqrt (* (fma (* (/ l Om) l) -2.0 t) t_2))
       (if (<= t_1 INFINITY)
         (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
         (sqrt (* (/ (* U* (* (* l l) n)) (* Om Om)) t_2)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = (n + n) * U;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((t * (n + n))) * sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((fma(((l / Om) * l), -2.0, t) * t_2));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((U_42_ * ((l * l) * n)) / (Om * Om)) * t_2));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	t_2 = Float64(Float64(n + n) * U)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * t_2));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * Om)) * t_2));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, Block[{t$95$2 = N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 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[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 31.6%

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

   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 35.9

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

   4. Applied rewrites 35.9%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 25.0

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

   6. Applied rewrites 25.0%

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

Alternative 13: 52.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(\left(\left(-n\right) \cdot \sqrt{2}\right) \cdot \ell\right) \cdot \sqrt{U* \cdot U}}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* t (+ n n))) (sqrt U))
     (if (<= t_1 5e+153)
       (sqrt (* (fma (* (/ l Om) l) -2.0 t) (* (+ n n) U)))
       (if (<= t_1 INFINITY)
         (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
         (- (/ (* (* (* (- n) (sqrt 2.0)) l) (sqrt (* U* U))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((fma(((l / Om) * l), -2.0, t) * ((n + n) * U)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = -((((-n * sqrt(2.0)) * l) * sqrt((U_42_ * U))) / Om);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(l / Om) * l), -2.0, t) * Float64(Float64(n + n) * U)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(-n) * sqrt(2.0)) * l) * sqrt(Float64(U_42_ * U))) / Om));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 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[(N[(N[(N[((-n) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision])]]]]

Derivation

1. Split input into 4 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 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 31.6%

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

   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 35.9

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

   4. Applied rewrites 35.9%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.0

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

   6. Applied rewrites 5.0%

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

   7. Taylor expanded in U* around -inf

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-+.f64 N/A

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

   9. Applied rewrites 22.3%

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

Alternative 14: 52.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

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

   1. Initial program 69.0%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.5

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

   6. Applied rewrites 72.5%

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

   7. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.8

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

   9. Applied rewrites 63.8%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.0

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

   6. Applied rewrites 5.0%

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

   7. Taylor expanded in U* around -inf

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-+.f64 N/A

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

   9. Applied rewrites 22.3%

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

Alternative 15: 44.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(\left(\left(-n\right) \cdot \sqrt{2}\right) \cdot \ell\right) \cdot \sqrt{U* \cdot U}}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* t (+ n n))) (sqrt U))
     (if (<= t_1 5e+153)
       (sqrt (* t (* (+ n n) U)))
       (- (/ (* (* (* (- n) (sqrt 2.0)) l) (sqrt (* U* U))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = -((((-n * sqrt(2.0)) * l) * sqrt((U_42_ * U))) / Om);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((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((t * (n + n))) * sqrt(u)
    else if (t_1 <= 5d+153) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = -((((-n * sqrt(2.0d0)) * l) * sqrt((u_42 * u))) / om)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((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((t * (n + n))) * Math.sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = -((((-n * Math.sqrt(2.0)) * l) * Math.sqrt((U_42_ * U))) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((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((t * (n + n))) * math.sqrt(U)
	elif t_1 <= 5e+153:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = -((((-n * math.sqrt(2.0)) * l) * math.sqrt((U_42_ * U))) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(-n) * sqrt(2.0)) * l) * sqrt(Float64(U_42_ * U))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	elseif (t_1 <= 5e+153)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = -((((-n * sqrt(2.0)) * l) * sqrt((U_42_ * U))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[(N[(N[(N[((-n) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 76.0%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 20.2%

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.2

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

   6. Applied rewrites 29.2%

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

   7. Taylor expanded in U* around -inf

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-+.f64 N/A

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

   9. Applied rewrites 21.5%

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

Alternative 16: 44.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* t (+ n n))) (sqrt U))
     (if (<= t_1 5e+153)
       (sqrt (* t (* (+ n n) U)))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((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((t * (n + n))) * sqrt(u)
    else if (t_1 <= 5d+153) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = sqrt((u_42 * u)) * (((sqrt(2.0d0) * n) * l) / om)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((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((t * (n + n))) * Math.sqrt(U);
	} else if (t_1 <= 5e+153) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((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((t * (n + n))) * math.sqrt(U)
	elif t_1 <= 5e+153:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((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((t * (n + n))) * sqrt(U);
	elseif (t_1 <= 5e+153)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   3. Applied rewrites 10.5%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 10.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.1

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

   10. Applied rewrites 31.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 76.0%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 20.2%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 21.0

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

   4. Applied rewrites 21.0%

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

Alternative 17: 40.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.8e+76)
   (sqrt (* t (* (+ n n) U)))
   (sqrt (* -4.0 (/ (* U (* (* l l) n)) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.8e+76) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 3.8d+76) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = sqrt(((-4.0d0) * ((u * ((l * l) * n)) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.8e+76) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = Math.sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.8e+76:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = math.sqrt((-4.0 * ((U * ((l * l) * n)) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.8e+76)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(Float64(l * l) * n)) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.8e+76)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.8e+76], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.80000000000000024e76

   1. Initial program 54.5%

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

   3. Applied rewrites 55.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 41.4%

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

   if 3.80000000000000024e76 < l

   1. Initial program 27.1%

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

   2. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 22.9

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

   7. Applied rewrites 22.9%

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

Alternative 18: 39.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n + n\right) \cdot U\\ \mathbf{if}\;t \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (+ n n) U)))
   (if (<= t 3e-309) (sqrt (* t t_1)) (* (sqrt t) (sqrt t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n + n) * U;
	double tmp;
	if (t <= 3e-309) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = sqrt(t) * sqrt(t_1);
	}
	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 = (n + n) * u
    if (t <= 3d-309) then
        tmp = sqrt((t * t_1))
    else
        tmp = sqrt(t) * sqrt(t_1)
    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 = (n + n) * U;
	double tmp;
	if (t <= 3e-309) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt(t_1);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n + n) * U
	tmp = 0
	if t <= 3e-309:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.sqrt(t) * math.sqrt(t_1)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n + n) * U)
	tmp = 0.0
	if (t <= 3e-309)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = Float64(sqrt(t) * sqrt(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n + n) * U;
	tmp = 0.0;
	if (t <= 3e-309)
		tmp = sqrt((t * t_1));
	else
		tmp = sqrt(t) * sqrt(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[t, 3e-309], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.000000000000001e-309

   1. Initial program 50.2%

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

   3. Applied rewrites 53.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 36.5%

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

   if 3.000000000000001e-309 < t

   1. Initial program 49.2%

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

   3. Applied rewrites 52.3%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 35.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 42.2

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

   8. Applied rewrites 42.2%

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

Alternative 19: 39.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot \left(n + n\right)\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 2e-116)
     (sqrt (* (+ n n) (* U t)))
     (if (<= t_1 5e+153)
       (sqrt (* t (* (+ n n) U)))
       (sqrt (* (* t (+ n n)) U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 2e-116) {
		tmp = sqrt(((n + n) * (U * t)));
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt(((t * (n + n)) * U));
	}
	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 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 2d-116) then
        tmp = sqrt(((n + n) * (u * t)))
    else if (t_1 <= 5d+153) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = sqrt(((t * (n + n)) * u))
    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 = Math.sqrt((((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 <= 2e-116) {
		tmp = Math.sqrt(((n + n) * (U * t)));
	} else if (t_1 <= 5e+153) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = Math.sqrt(((t * (n + n)) * U));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((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 <= 2e-116:
		tmp = math.sqrt(((n + n) * (U * t)))
	elif t_1 <= 5e+153:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = math.sqrt(((t * (n + n)) * U))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	tmp = 0.0
	if (t_1 <= 2e-116)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(t * Float64(n + n)) * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_1 <= 2e-116)
		tmp = sqrt(((n + n) * (U * t)));
	elseif (t_1 <= 5e+153)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = sqrt(((t * (n + n)) * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, If[LessEqual[t$95$1, 2e-116], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2e-116

   1. Initial program 26.5%

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 38.2%

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

   if 2e-116 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 98.0%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 76.5%

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

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 20.2%

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

   3. Applied rewrites 28.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 9.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 11.5

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

   8. Applied rewrites 11.5%

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

Alternative 20: 38.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left(t \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (* (* t (+ n n)) U))))
   (if (<= t_1 0.0) t_2 (if (<= t_1 5e+153) (sqrt (* t (* (+ n n) U))) t_2))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt(((t * (n + n)) * U));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+153) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    t_2 = sqrt(((t * (n + n)) * u))
    if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 5d+153) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = t_2
    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 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = Math.sqrt(((t * (n + n)) * U));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+153) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	t_2 = math.sqrt(((t * (n + n)) * U))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 5e+153:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = t_2
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_)))))
	t_2 = sqrt(Float64(Float64(t * Float64(n + n)) * U))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+153)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	t_2 = sqrt(((t * (n + n)) * U));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+153)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, Block[{t$95$2 = N[Sqrt[N[(N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+153], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

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 or 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 18.1%

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 9.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 16.3

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

   8. Applied rewrites 16.3%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 97.3%

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 76.0%

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

Alternative 21: 36.2% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Applied rewrites 52.6%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 36.2%

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

Reproduce

herbie shell --seed 2025114 
(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*))))))