Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 58.3%

Time: 8.4s

Alternatives: 14

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 58.3% accurate, 0.3× speedup

Math

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

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)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (if (<= t_1 INFINITY)
       t_1
       (sqrt
        (*
         -2.0
         (*
          U
          (*
           (pow l 2.0)
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow 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 = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = sqrt((-2.0 * (U * (pow(l, 2.0) * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(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(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64((l ^ 2.0) * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (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[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[Sqrt[N[(-2.0 * N[(U * N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   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 50.5%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-pow.f64 20.0

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

   4. Applied rewrites 20.0%

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

Alternative 2: 57.7% accurate, 0.3× speedup

Math

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

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)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (if (<= t_1 INFINITY)
       t_1
       (sqrt
        (*
         (pow l 2.0)
         (*
          n
          (fma
           -4.0
           (/ U Om)
           (* -2.0 (/ (* U (* n (- U U*))) (pow 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 = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = sqrt((pow(l, 2.0) * (n * fma(-4.0, (U / Om), (-2.0 * ((U * (n * (U - U_42_))) / pow(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(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = sqrt(Float64((l ^ 2.0) * Float64(n * fma(-4.0, Float64(U / Om), Float64(-2.0 * Float64(Float64(U * Float64(n * Float64(U - U_42_))) / (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[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[Sqrt[N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(-4.0 * N[(U / Om), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   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 50.5%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 19.1

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

   7. Applied rewrites 19.1%

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

Alternative 3: 57.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

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

   1. Initial program 50.5%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 16.6

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

   7. Applied rewrites 16.6%

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

Alternative 4: 52.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{-154}:\\ \;\;\;\;U \cdot \left(\sqrt{\frac{n \cdot t}{U}} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        5e-154)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (sqrt
      (*
       t_1
       (+ t (* -1.0 (/ (* (pow l 2.0) (+ 2.0 (/ (* n (- U U*)) Om))) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 5e-154) {
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else {
		tmp = sqrt((t_1 * (t + (-1.0 * ((pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / 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 = (2.0d0 * n) * u
    if (sqrt((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 5d-154) then
        tmp = u * (sqrt(((n * t) / u)) * sqrt(2.0d0))
    else
        tmp = sqrt((t_1 * (t + ((-1.0d0) * (((l ** 2.0d0) * (2.0d0 + ((n * (u - u_42)) / om))) / 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 = (2.0 * n) * U;
	double tmp;
	if (Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 5e-154) {
		tmp = U * (Math.sqrt(((n * t) / U)) * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt((t_1 * (t + (-1.0 * ((Math.pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 5e-154:
		tmp = U * (math.sqrt(((n * t) / U)) * math.sqrt(2.0))
	else:
		tmp = math.sqrt((t_1 * (t + (-1.0 * ((math.pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 5e-154)
		tmp = Float64(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	else
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(-1.0 * Float64(Float64((l ^ 2.0) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 5e-154)
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	else
		tmp = sqrt((t_1 * (t + (-1.0 * (((l ^ 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * 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], 5e-154], N[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t + N[(-1.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.0000000000000002e-154

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   if 5.0000000000000002e-154 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

Alternative 5: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \left(\sqrt{\frac{n \cdot t}{U}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(2 + \frac{U \cdot n}{Om}\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + -1 \cdot \frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (if (<= t_2 5e+229)
       (sqrt
        (* t_1 (+ t (* -1.0 (/ (* (pow l 2.0) (+ 2.0 (/ (* U n) Om))) Om)))))
       (sqrt
        (*
         t_1
         (+ t (* -1.0 (/ (/ (* (pow l 2.0) (* n (- U U*))) Om) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else if (t_2 <= 5e+229) {
		tmp = sqrt((t_1 * (t + (-1.0 * ((pow(l, 2.0) * (2.0 + ((U * n) / Om))) / Om)))));
	} else {
		tmp = sqrt((t_1 * (t + (-1.0 * (((pow(l, 2.0) * (n * (U - U_42_))) / Om) / 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_2 <= 0.0d0) then
        tmp = u * (sqrt(((n * t) / u)) * sqrt(2.0d0))
    else if (t_2 <= 5d+229) then
        tmp = sqrt((t_1 * (t + ((-1.0d0) * (((l ** 2.0d0) * (2.0d0 + ((u * n) / om))) / om)))))
    else
        tmp = sqrt((t_1 * (t + ((-1.0d0) * ((((l ** 2.0d0) * (n * (u - u_42))) / om) / 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 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (Math.sqrt(((n * t) / U)) * Math.sqrt(2.0));
	} else if (t_2 <= 5e+229) {
		tmp = Math.sqrt((t_1 * (t + (-1.0 * ((Math.pow(l, 2.0) * (2.0 + ((U * n) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((t_1 * (t + (-1.0 * (((Math.pow(l, 2.0) * (n * (U - U_42_))) / Om) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * (math.sqrt(((n * t) / U)) * math.sqrt(2.0))
	elif t_2 <= 5e+229:
		tmp = math.sqrt((t_1 * (t + (-1.0 * ((math.pow(l, 2.0) * (2.0 + ((U * n) / Om))) / Om)))))
	else:
		tmp = math.sqrt((t_1 * (t + (-1.0 * (((math.pow(l, 2.0) * (n * (U - U_42_))) / Om) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	elseif (t_2 <= 5e+229)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(-1.0 * Float64(Float64((l ^ 2.0) * Float64(2.0 + Float64(Float64(U * n) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(-1.0 * Float64(Float64(Float64((l ^ 2.0) * Float64(n * Float64(U - U_42_))) / Om) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	elseif (t_2 <= 5e+229)
		tmp = sqrt((t_1 * (t + (-1.0 * (((l ^ 2.0) * (2.0 + ((U * n) / Om))) / Om)))));
	else
		tmp = sqrt((t_1 * (t + (-1.0 * ((((l ^ 2.0) * (n * (U - U_42_))) / Om) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+229], N[Sqrt[N[(t$95$1 * N[(t + N[(-1.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 + N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t + N[(-1.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000005e229

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in U around inf

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

      1. lower-*.f64 38.1

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

   10. Applied rewrites 38.1%

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

   if 5.0000000000000005e229 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 45.1

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

   10. Applied rewrites 45.1%

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

Alternative 6: 48.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{{\ell}^{2}}{Om}\\ \mathbf{if}\;Om \leq -37000000:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_1\right)\right)}\\ \mathbf{elif}\;Om \leq 1.85 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -1 \cdot \frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (pow l 2.0) Om)))))
   (if (<= Om -37000000.0)
     (sqrt (* 2.0 (* U (* n t_1))))
     (if (<= Om 1.85e-121)
       (sqrt
        (*
         (* (* 2.0 n) U)
         (+ t (* -1.0 (/ (/ (* (pow l 2.0) (* n (- U U*))) Om) Om)))))
       (sqrt (* n (* 2.0 (* U t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (2.0 * (pow(l, 2.0) / Om));
	double tmp;
	if (Om <= -37000000.0) {
		tmp = sqrt((2.0 * (U * (n * t_1))));
	} else if (Om <= 1.85e-121) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-1.0 * (((pow(l, 2.0) * (n * (U - U_42_))) / Om) / Om)))));
	} else {
		tmp = sqrt((n * (2.0 * (U * 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 = t - (2.0d0 * ((l ** 2.0d0) / om))
    if (om <= (-37000000.0d0)) then
        tmp = sqrt((2.0d0 * (u * (n * t_1))))
    else if (om <= 1.85d-121) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-1.0d0) * ((((l ** 2.0d0) * (n * (u - u_42))) / om) / om)))))
    else
        tmp = sqrt((n * (2.0d0 * (u * 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 = t - (2.0 * (Math.pow(l, 2.0) / Om));
	double tmp;
	if (Om <= -37000000.0) {
		tmp = Math.sqrt((2.0 * (U * (n * t_1))));
	} else if (Om <= 1.85e-121) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-1.0 * (((Math.pow(l, 2.0) * (n * (U - U_42_))) / Om) / Om)))));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * t_1))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (2.0 * (math.pow(l, 2.0) / Om))
	tmp = 0
	if Om <= -37000000.0:
		tmp = math.sqrt((2.0 * (U * (n * t_1))))
	elif Om <= 1.85e-121:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-1.0 * (((math.pow(l, 2.0) * (n * (U - U_42_))) / Om) / Om)))))
	else:
		tmp = math.sqrt((n * (2.0 * (U * t_1))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))
	tmp = 0.0
	if (Om <= -37000000.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_1))));
	elseif (Om <= 1.85e-121)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-1.0 * Float64(Float64(Float64((l ^ 2.0) * Float64(n * Float64(U - U_42_))) / Om) / Om)))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - (2.0 * ((l ^ 2.0) / Om));
	tmp = 0.0;
	if (Om <= -37000000.0)
		tmp = sqrt((2.0 * (U * (n * t_1))));
	elseif (Om <= 1.85e-121)
		tmp = sqrt((((2.0 * n) * U) * (t + (-1.0 * ((((l ^ 2.0) * (n * (U - U_42_))) / Om) / Om)))));
	else
		tmp = sqrt((n * (2.0 * (U * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -37000000.0], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.85e-121], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-1.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -3.7e7

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.1

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

   4. Applied rewrites 44.1%

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

   if -3.7e7 < Om < 1.8500000000000001e-121

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 45.1

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

   10. Applied rewrites 45.1%

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

   if 1.8500000000000001e-121 < Om

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in n around 0

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 43.6

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

   7. Applied rewrites 43.6%

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

Alternative 7: 48.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \left(\sqrt{\frac{n \cdot t}{U}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (if (<= t_2 2e+302)
       (sqrt (* t_1 (- t (* 2.0 (/ (pow l 2.0) Om)))))
       (* (/ (* l (* n (sqrt 2.0))) Om) (sqrt (* U U*)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else if (t_2 <= 2e+302) {
		tmp = sqrt((t_1 * (t - (2.0 * (pow(l, 2.0) / Om)))));
	} else {
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	}
	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 = (2.0d0 * n) * u
    t_2 = t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_2 <= 0.0d0) then
        tmp = u * (sqrt(((n * t) / u)) * sqrt(2.0d0))
    else if (t_2 <= 2d+302) then
        tmp = sqrt((t_1 * (t - (2.0d0 * ((l ** 2.0d0) / om)))))
    else
        tmp = ((l * (n * sqrt(2.0d0))) / om) * sqrt((u * u_42))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (Math.sqrt(((n * t) / U)) * Math.sqrt(2.0));
	} else if (t_2 <= 2e+302) {
		tmp = Math.sqrt((t_1 * (t - (2.0 * (Math.pow(l, 2.0) / Om)))));
	} else {
		tmp = ((l * (n * Math.sqrt(2.0))) / Om) * Math.sqrt((U * U_42_));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * (math.sqrt(((n * t) / U)) * math.sqrt(2.0))
	elif t_2 <= 2e+302:
		tmp = math.sqrt((t_1 * (t - (2.0 * (math.pow(l, 2.0) / Om)))))
	else:
		tmp = ((l * (n * math.sqrt(2.0))) / Om) * math.sqrt((U * U_42_))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	elseif (t_2 <= 2e+302)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))));
	else
		tmp = Float64(Float64(Float64(l * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	elseif (t_2 <= 2e+302)
		tmp = sqrt((t_1 * (t - (2.0 * ((l ^ 2.0) / Om)))));
	else
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+302], N[Sqrt[N[(t$95$1 * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000002e302

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.8

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

   4. Applied rewrites 43.8%

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

   if 2.0000000000000002e302 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

Alternative 8: 46.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{{\ell}^{2}}{Om}\\ \mathbf{if}\;Om \leq -1.7 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_1\right)\right)}\\ \mathbf{elif}\;Om \leq 6.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (pow l 2.0) Om)))))
   (if (<= Om -1.7e-305)
     (sqrt (* 2.0 (* U (* n t_1))))
     (if (<= Om 6.8e-127)
       (* (/ (* l (* n (sqrt 2.0))) Om) (sqrt (* U U*)))
       (sqrt (* n (* 2.0 (* U t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (2.0 * (pow(l, 2.0) / Om));
	double tmp;
	if (Om <= -1.7e-305) {
		tmp = sqrt((2.0 * (U * (n * t_1))));
	} else if (Om <= 6.8e-127) {
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	} else {
		tmp = sqrt((n * (2.0 * (U * 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 = t - (2.0d0 * ((l ** 2.0d0) / om))
    if (om <= (-1.7d-305)) then
        tmp = sqrt((2.0d0 * (u * (n * t_1))))
    else if (om <= 6.8d-127) then
        tmp = ((l * (n * sqrt(2.0d0))) / om) * sqrt((u * u_42))
    else
        tmp = sqrt((n * (2.0d0 * (u * 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 = t - (2.0 * (Math.pow(l, 2.0) / Om));
	double tmp;
	if (Om <= -1.7e-305) {
		tmp = Math.sqrt((2.0 * (U * (n * t_1))));
	} else if (Om <= 6.8e-127) {
		tmp = ((l * (n * Math.sqrt(2.0))) / Om) * Math.sqrt((U * U_42_));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * t_1))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (2.0 * (math.pow(l, 2.0) / Om))
	tmp = 0
	if Om <= -1.7e-305:
		tmp = math.sqrt((2.0 * (U * (n * t_1))))
	elif Om <= 6.8e-127:
		tmp = ((l * (n * math.sqrt(2.0))) / Om) * math.sqrt((U * U_42_))
	else:
		tmp = math.sqrt((n * (2.0 * (U * t_1))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))
	tmp = 0.0
	if (Om <= -1.7e-305)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_1))));
	elseif (Om <= 6.8e-127)
		tmp = Float64(Float64(Float64(l * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U * U_42_)));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - (2.0 * ((l ^ 2.0) / Om));
	tmp = 0.0;
	if (Om <= -1.7e-305)
		tmp = sqrt((2.0 * (U * (n * t_1))));
	elseif (Om <= 6.8e-127)
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	else
		tmp = sqrt((n * (2.0 * (U * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.7e-305], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 6.8e-127], N[(N[(N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.7e-305

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.1

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

   4. Applied rewrites 44.1%

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

   if -1.7e-305 < Om < 6.7999999999999997e-127

   1. Initial program 50.5%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

   if 6.7999999999999997e-127 < Om

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in n around 0

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 43.6

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

   7. Applied rewrites 43.6%

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

Alternative 9: 44.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 44.1

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

   4. Applied rewrites 44.1%

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

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

   1. Initial program 50.5%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

Alternative 10: 43.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \left(\sqrt{\frac{n \cdot t}{U}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (if (<= t_2 2e+302)
       (sqrt (* t_1 (* t 1.0)))
       (* (/ (* l (* n (sqrt 2.0))) Om) (sqrt (* U U*)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else if (t_2 <= 2e+302) {
		tmp = sqrt((t_1 * (t * 1.0)));
	} else {
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	}
	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 = (2.0d0 * n) * u
    t_2 = t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_2 <= 0.0d0) then
        tmp = u * (sqrt(((n * t) / u)) * sqrt(2.0d0))
    else if (t_2 <= 2d+302) then
        tmp = sqrt((t_1 * (t * 1.0d0)))
    else
        tmp = ((l * (n * sqrt(2.0d0))) / om) * sqrt((u * u_42))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (Math.sqrt(((n * t) / U)) * Math.sqrt(2.0));
	} else if (t_2 <= 2e+302) {
		tmp = Math.sqrt((t_1 * (t * 1.0)));
	} else {
		tmp = ((l * (n * Math.sqrt(2.0))) / Om) * Math.sqrt((U * U_42_));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * (math.sqrt(((n * t) / U)) * math.sqrt(2.0))
	elif t_2 <= 2e+302:
		tmp = math.sqrt((t_1 * (t * 1.0)))
	else:
		tmp = ((l * (n * math.sqrt(2.0))) / Om) * math.sqrt((U * U_42_))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	elseif (t_2 <= 2e+302)
		tmp = sqrt(Float64(t_1 * Float64(t * 1.0)));
	else
		tmp = Float64(Float64(Float64(l * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	elseif (t_2 <= 2e+302)
		tmp = sqrt((t_1 * (t * 1.0)));
	else
		tmp = ((l * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+302], N[Sqrt[N[(t$95$1 * N[(t * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000002e302

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 36.5%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t \cdot 1\right)} \]

   if 2.0000000000000002e302 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

Alternative 11: 39.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \left(\sqrt{\frac{n \cdot t}{U}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+266}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* U (* (sqrt (/ (* n t) U)) (sqrt 2.0)))
     (if (<= t_2 1e+266)
       (sqrt (* t_1 (* t 1.0)))
       (sqrt (* 2.0 (* U (* n t))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	} else if (t_2 <= 1e+266) {
		tmp = sqrt((t_1 * (t * 1.0)));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_2 <= 0.0d0) then
        tmp = u * (sqrt(((n * t) / u)) * sqrt(2.0d0))
    else if (t_2 <= 1d+266) then
        tmp = sqrt((t_1 * (t * 1.0d0)))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * (Math.sqrt(((n * t) / U)) * Math.sqrt(2.0));
	} else if (t_2 <= 1e+266) {
		tmp = Math.sqrt((t_1 * (t * 1.0)));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * (math.sqrt(((n * t) / U)) * math.sqrt(2.0))
	elif t_2 <= 1e+266:
		tmp = math.sqrt((t_1 * (t * 1.0)))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * Float64(sqrt(Float64(Float64(n * t) / U)) * sqrt(2.0)));
	elseif (t_2 <= 1e+266)
		tmp = sqrt(Float64(t_1 * Float64(t * 1.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * (sqrt(((n * t) / U)) * sqrt(2.0));
	elseif (t_2 <= 1e+266)
		tmp = sqrt((t_1 * (t * 1.0)));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(U * N[(N[Sqrt[N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+266], N[Sqrt[N[(t$95$1 * N[(t * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.2%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 15.0%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 18.6

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

   13. Applied rewrites 18.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e266

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 36.5%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t \cdot 1\right)} \]

   if 1e266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.1

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

   4. Applied rewrites 36.1%

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

Alternative 12: 39.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+266}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt (* n (* 2.0 (* U t))))
     (if (<= t_2 1e+266)
       (sqrt (* t_1 (* t 1.0)))
       (sqrt (* 2.0 (* U (* n t))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * (2.0 * (U * t))));
	} else if (t_2 <= 1e+266) {
		tmp = sqrt((t_1 * (t * 1.0)));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_2 <= 0.0d0) then
        tmp = sqrt((n * (2.0d0 * (u * t))))
    else if (t_2 <= 1d+266) then
        tmp = sqrt((t_1 * (t * 1.0d0)))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((n * (2.0 * (U * t))));
	} else if (t_2 <= 1e+266) {
		tmp = Math.sqrt((t_1 * (t * 1.0)));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((n * (2.0 * (U * t))))
	elif t_2 <= 1e+266:
		tmp = math.sqrt((t_1 * (t * 1.0)))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * t))));
	elseif (t_2 <= 1e+266)
		tmp = sqrt(Float64(t_1 * Float64(t * 1.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((n * (2.0 * (U * t))));
	elseif (t_2 <= 1e+266)
		tmp = sqrt((t_1 * (t * 1.0)));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+266], N[Sqrt[N[(t$95$1 * N[(t * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 35.6

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

   7. Applied rewrites 35.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e266

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 36.5%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t \cdot 1\right)} \]

   if 1e266 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.1

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

   4. Applied rewrites 36.1%

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

Alternative 13: 36.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 8.2 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 8.2e+24)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* n (* 2.0 (* U t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 8.2e+24) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((n * (2.0 * (U * t))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 8.2d+24) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((n * (2.0d0 * (u * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 8.2e+24) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((n * (2.0 * (U * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 8.2e+24:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((n * (2.0 * (U * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 8.2e+24)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(n * Float64(2.0 * Float64(U * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= 8.2e+24)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((n * (2.0 * (U * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 8.2e+24], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 8.2000000000000002e24

   1. Initial program 50.5%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.1

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

   4. Applied rewrites 36.1%

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

   if 8.2000000000000002e24 < Om

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 35.6

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

   7. Applied rewrites 35.6%

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

Alternative 14: 36.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
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 * (U * (n * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 36.1

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

4. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2025134 
(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*))))))