Toniolo and Linder, Equation (13)

Percentage Accurate: 49.1% → 62.6%

Time: 9.6s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.1% 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: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := n \cdot \left(U - U*\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -1 \cdot \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \frac{\left(\ell \cdot \ell\right) \cdot t\_2}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{2 + \frac{t\_2}{Om}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

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*))))))
        (t_2 (* n (- U U*))))
   (if (<= t_1 0.0)
     (sqrt
      (*
       (+ n n)
       (* U (+ t (* -1.0 (/ (fma 2.0 (* l l) (/ (* (* l l) t_2) Om)) Om))))))
     (if (<= t_1 4e+147)
       t_1
       (if (<= t_1 INFINITY)
         (sqrt
          (*
           (+ n n)
           (*
            U
            (-
             (fma -2.0 (* l (/ l Om)) t)
             (* n (* (* (/ l Om) (/ l Om)) (- U U*)))))))
         (sqrt (* -2.0 (* U (* (* l l) (* n (/ (+ 2.0 (/ t_2 Om)) Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = n * (U - U_42_);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((n + n) * (U * (t + (-1.0 * (fma(2.0, (l * l), (((l * l) * t_2) / Om)) / Om))))));
	} else if (t_1 <= 4e+147) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((n + n) * (U * (fma(-2.0, (l * (l / Om)), t) - (n * (((l / Om) * (l / Om)) * (U - U_42_)))))));
	} else {
		tmp = sqrt((-2.0 * (U * ((l * l) * (n * ((2.0 + (t_2 / Om)) / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = Float64(n * Float64(U - U_42_))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t + Float64(-1.0 * Float64(fma(2.0, Float64(l * l), Float64(Float64(Float64(l * l) * t_2) / Om)) / Om))))));
	elseif (t_1 <= 4e+147)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(Float64(2.0 + Float64(t_2 / Om)) / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t + N[(-1.0 * N[(N[(2.0 * N[(l * l), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 4e+147], t$95$1, If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(2.0 + N[(t$95$2 / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.3%

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

   3. Applied rewrites 34.9%

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

   4. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 35.7

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

   6. Applied rewrites 35.7%

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

   if 0.0 < (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*))))) < 3.9999999999999999e147

   1. Initial program 97.6%

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

   if 3.9999999999999999e147 < (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 33.0%

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

   3. Applied rewrites 43.6%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 9.1%

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

   4. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. 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. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 32.9

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

   6. Applied rewrites 32.9%

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

   7. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 40.5

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

   9. Applied rewrites 40.5%

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

Alternative 2: 61.4% accurate, 0.3× speedup

Math

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

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*))))))
        (t_2 (* n (- U U*))))
   (if (<= t_1 0.0)
     (sqrt
      (*
       (+ n n)
       (* U (+ t (* -1.0 (/ (fma 2.0 (* l l) (/ (* (* l l) t_2) Om)) Om))))))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (-
          (fma -2.0 (* l (/ l Om)) t)
          (* n (* (* (/ l Om) (/ l Om)) (- U U*))))
         (* (+ n n) U)))
       (sqrt (* -2.0 (* U (* (* l l) (* n (/ (+ 2.0 (/ t_2 Om)) Om))))))))))

C

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

Julia

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t + N[(-1.0 * N[(N[(2.0 * N[(l * l), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(2.0 + N[(t$95$2 / Om), $MachinePrecision]), $MachinePrecision] / Om), $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 11.3%

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

   3. Applied rewrites 34.9%

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

   4. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 35.7

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

   6. Applied rewrites 35.7%

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

   if 0.0 < (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 68.3%

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

   3. Applied rewrites 71.5%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 9.1%

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

   4. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. 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. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 32.9

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

   6. Applied rewrites 32.9%

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

   7. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 40.5

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

   9. Applied rewrites 40.5%

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

Alternative 3: 56.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U - U*\right)\\ t_2 := 2 + \frac{t\_1}{Om}\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t\_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -1 \cdot \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \frac{\left(\ell \cdot \ell\right) \cdot t\_1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(\left(-\frac{\left(\ell \cdot \ell\right) \cdot t\_2}{Om}\right) + t\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{t\_2}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- U U*)))
        (t_2 (+ 2.0 (/ t_1 Om)))
        (t_3 (* (* 2.0 n) U))
        (t_4
         (sqrt
          (*
           t_3
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_4 0.0)
     (sqrt
      (*
       (+ n n)
       (* U (+ t (* -1.0 (/ (fma 2.0 (* l l) (/ (* (* l l) t_1) Om)) Om))))))
     (if (<= t_4 4e+73)
       (sqrt (* t_3 (+ (- (/ (* (* l l) t_2) Om)) t)))
       (if (<= t_4 INFINITY)
         (sqrt (* t_3 (fma -2.0 (* l (/ l Om)) t)))
         (sqrt (* -2.0 (* U (* (* l l) (* n (/ t_2 Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U - U_42_);
	double t_2 = 2.0 + (t_1 / Om);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((n + n) * (U * (t + (-1.0 * (fma(2.0, (l * l), (((l * l) * t_1) / Om)) / Om))))));
	} else if (t_4 <= 4e+73) {
		tmp = sqrt((t_3 * (-(((l * l) * t_2) / Om) + t)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * fma(-2.0, (l * (l / Om)), t)));
	} else {
		tmp = sqrt((-2.0 * (U * ((l * l) * (n * (t_2 / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U - U_42_))
	t_2 = Float64(2.0 + Float64(t_1 / Om))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * 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_4 <= 0.0)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t + Float64(-1.0 * Float64(fma(2.0, Float64(l * l), Float64(Float64(Float64(l * l) * t_1) / Om)) / Om))))));
	elseif (t_4 <= 4e+73)
		tmp = sqrt(Float64(t_3 * Float64(Float64(-Float64(Float64(Float64(l * l) * t_2) / Om)) + t)));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(t_2 / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * 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$4, 0.0], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t + N[(-1.0 * N[(N[(2.0 * N[(l * l), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 4e+73], N[Sqrt[N[(t$95$3 * N[((-N[(N[(N[(l * l), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision]) + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(t$95$2 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.3%

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

   3. Applied rewrites 34.9%

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

   4. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 35.7

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

   6. Applied rewrites 35.7%

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

   if 0.0 < (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*))))) < 3.99999999999999993e73

   1. Initial program 97.2%

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

   2. Taylor expanded in 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 3.99999999999999993e73 < (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 46.8%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 45.0

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

   4. Applied rewrites 45.0%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 9.1%

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

   4. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. 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. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 32.9

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

   6. Applied rewrites 32.9%

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

   7. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 40.5

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

   9. Applied rewrites 40.5%

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

Alternative 4: 55.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := 2 + \frac{n \cdot \left(U - U*\right)}{Om}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(-\frac{\left(\ell \cdot \ell\right) \cdot t\_4}{Om}\right) + t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{t\_4}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        (t_4 (+ 2.0 (/ (* n (- U U*)) Om))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* t_1 n) U) 2.0))
     (if (<= t_3 5e+146)
       (sqrt (* t_2 (+ (- (/ (* (* l l) t_4) Om)) t)))
       (if (<= t_3 INFINITY)
         (sqrt (* t_2 t_1))
         (sqrt (* -2.0 (* U (* (* l l) (* n (/ t_4 Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(-2.0, (l * (l / Om)), t);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double t_4 = 2.0 + ((n * (U - U_42_)) / Om);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else if (t_3 <= 5e+146) {
		tmp = sqrt((t_2 * (-(((l * l) * t_4) / Om) + t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * t_1));
	} else {
		tmp = sqrt((-2.0 * (U * ((l * l) * (n * (t_4 / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= 5e+146)
		tmp = sqrt(Float64(t_2 * Float64(Float64(-Float64(Float64(Float64(l * l) * t_4) / Om)) + t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(t_4 / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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]}, Block[{t$95$4 = N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+146], N[Sqrt[N[(t$95$2 * N[((-N[(N[(N[(l * l), $MachinePrecision] * t$95$4), $MachinePrecision] / Om), $MachinePrecision]) + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(t$95$4 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.4

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

   4. Applied rewrites 36.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999999e146

   1. Initial program 97.2%

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

   2. Taylor expanded in 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 4.9999999999999999e146 < (*.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 46.8%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 45.0

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

   4. Applied rewrites 45.0%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 6.8%

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

   4. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. 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. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 35.3

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

   6. Applied rewrites 35.3%

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

   7. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 42.7

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

   9. Applied rewrites 42.7%

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

Alternative 5: 55.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* t_1 n) U) 2.0))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 t_1))
       (sqrt
        (*
         -2.0
         (* U (* (* l l) (* n (/ (+ 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 = fma(-2.0, (l * (l / Om)), t);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * t_1));
	} else {
		tmp = sqrt((-2.0 * (U * ((l * l) * (n * ((2.0 + ((n * (U - U_42_)) / Om)) / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / Om))))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 9.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.4

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

   4. Applied rewrites 36.4%

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

   if 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 68.3%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 62.8

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

   4. Applied rewrites 62.8%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 6.8%

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

   4. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. 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. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 35.3

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

   6. Applied rewrites 35.3%

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

   7. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 42.7

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

   9. Applied rewrites 42.7%

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

Alternative 6: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{if}\;n \leq -2.05 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (+ n n) (* U (- t (* n (* (* (/ l Om) (/ l Om)) (- U U*)))))))))
   (if (<= n -2.05e-112)
     t_1
     (if (<= n 3e-62)
       (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n + n) * (U * (t - (n * (((l / Om) * (l / Om)) * (U - U_42_)))))));
	double tmp;
	if (n <= -2.05e-112) {
		tmp = t_1;
	} else if (n <= 3e-62) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))))))
	tmp = 0.0
	if (n <= -2.05e-112)
		tmp = t_1;
	elseif (n <= 3e-62)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -2.05e-112], t$95$1, If[LessEqual[n, 3e-62], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.04999999999999998e-112 or 3.0000000000000001e-62 < n

   1. Initial program 54.0%

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

   3. Applied rewrites 58.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 58.6%

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

   if -2.04999999999999998e-112 < n < 3.0000000000000001e-62

   1. Initial program 42.1%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 50.6

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

   4. Applied rewrites 50.6%

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

Alternative 7: 53.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * Om)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * 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 9.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.4

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

   4. Applied rewrites 36.4%

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

   if 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 68.3%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 62.8

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

   4. Applied rewrites 62.8%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 6.8%

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

   4. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 30.4

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

   6. Applied rewrites 30.4%

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

Alternative 8: 53.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t + Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * Om))))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 9.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.4

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

   4. Applied rewrites 36.4%

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

   if 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 68.3%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 62.8

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

   4. Applied rewrites 62.8%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 6.8%

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

   4. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 12.6

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

   6. Applied rewrites 12.6%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 30.6

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

   9. Applied rewrites 30.6%

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

Alternative 9: 52.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.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 9.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.4

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

   4. Applied rewrites 36.4%

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

   if 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 68.3%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 62.8

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

   4. Applied rewrites 62.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 22.7

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

   4. Applied rewrites 22.7%

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

Alternative 10: 49.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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 57.7%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 54.1

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

   4. Applied rewrites 54.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 22.7

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

   4. Applied rewrites 22.7%

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

Alternative 11: 43.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 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 n) (* U t)))
     (if (<= t_2 5e+294)
       (sqrt (* t_1 t))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (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 + n) * (U * t)));
	} else if (t_2 <= 5e+294) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: 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 + n) * (u * t)))
    else if (t_2 <= 5d+294) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt((u_42 * u)) * (((sqrt(2.0d0) * n) * l) / om)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (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 + n) * (U * t)));
	} else if (t_2 <= 5e+294) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (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 + n) * (U * t)))
	elif t_2 <= 5e+294:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	elseif (t_2 <= 5e+294)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (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 + n) * (U * t)));
	elseif (t_2 <= 5e+294)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(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 + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+294], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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 9.9%

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

   3. Applied rewrites 34.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 30.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999999e294

   1. Initial program 97.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 75.0%

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

   if 4.9999999999999999e294 < (*.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 22.3%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 22.7

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

   4. Applied rewrites 22.7%

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

Alternative 12: 38.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.5e-167)
   (sqrt (* (* (* 2.0 n) U) t))
   (if (<= l 3.8e+39)
     (sqrt (* (* (* t n) U) 2.0))
     (sqrt (* -4.0 (/ (* U (* (* l l) n)) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.5e-167) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else if (l <= 3.8e+39) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 8.5d-167) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else if (l <= 3.8d+39) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((-4.0d0) * ((u * ((l * l) * n)) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.5e-167) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else if (l <= 3.8e+39) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.5e-167:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	elif l <= 3.8e+39:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((-4.0 * ((U * ((l * l) * n)) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.5e-167)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	elseif (l <= 3.8e+39)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(Float64(l * l) * n)) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8.5e-167)
		tmp = sqrt((((2.0 * n) * U) * t));
	elseif (l <= 3.8e+39)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.5e-167], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e+39], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.4999999999999994e-167

   1. Initial program 52.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 39.8%

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

   if 8.4999999999999994e-167 < l < 3.7999999999999998e39

   1. Initial program 58.2%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 43.2

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

   4. Applied rewrites 43.2%

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

   if 3.7999999999999998e39 < l

   1. Initial program 30.8%

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

   3. Applied rewrites 40.2%

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

   4. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. 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. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 34.3

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

   6. Applied rewrites 34.3%

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

   7. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 22.6

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

   9. Applied rewrites 22.6%

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

Alternative 13: 37.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2.9e-278)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* (* 2.0 n) U)) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.9e-278) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 2.9d-278) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.9e-278) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 2.9e-278:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2.9e-278)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 2.9e-278)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2.9e-278], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.9e-278

   1. Initial program 48.0%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.9

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

   4. Applied rewrites 33.9%

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

   if 2.9e-278 < t

   1. Initial program 50.2%

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

   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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 36.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-prod N/A

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

   9. Applied rewrites 42.9%

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

Alternative 14: 35.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.5e-167)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* (* (* t n) U) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.5e-167) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((((t * n) * U) * 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 8.5d-167) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((((t * n) * u) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.5e-167) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.5e-167:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.5e-167)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8.5e-167)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((((t * n) * U) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.5e-167], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.4999999999999994e-167

   1. Initial program 52.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 39.8%

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

   if 8.4999999999999994e-167 < l

   1. Initial program 43.6%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.4

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

   4. Applied rewrites 28.4%

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

Alternative 15: 35.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

2. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 35.5

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

4. Applied rewrites 35.5%

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

Alternative 16: 35.2% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Applied rewrites 52.5%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 35.2%

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

Reproduce

herbie shell --seed 2025101 
(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*))))))