Toniolo and Linder, Equation (13)

Percentage Accurate: 48.7% → 65.0%

Time: 8.1s

Alternatives: 14

Speedup: 1.7×

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: 48.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.0% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt
      (*
       (*
        (+
         t
         (*
          -1.0
          (/
           (fma
            -1.0
            (/ (* (pow l_m 2.0) (* n (- U* U))) Om)
            (* 2.0 (pow l_m 2.0)))
           Om)))
        (+ n n))
       U))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (- t (* 2.0 (/ l_m (/ Om l_m))))
          (* (* (* n (/ l_m Om)) (/ l_m Om)) (- U U*)))))
       (*
        l_m
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   6. Applied rewrites 43.9%

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

   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 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 52.8

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

   3. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

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

   1. Initial program 48.7%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 28.2

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

   4. Applied rewrites 28.2%

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

Alternative 2: 64.3% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (pow (* (* (* t n) U) 2.0) 0.5)
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (- t (* 2.0 (/ l_m (/ Om l_m))))
          (* (* (* n (/ l_m Om)) (/ l_m Om)) (- U U*)))))
       (*
        l_m
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = pow((((t * n) * U) * 2.0), 0.5);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((n + n) * U) * ((t - (2.0 * (l_m / (Om / l_m)))) - (((n * (l_m / Om)) * (l_m / Om)) * (U - U_42_)))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(t * n) * U) * 2.0) ^ 0.5;
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(Float64(t - Float64(2.0 * Float64(l_m / Float64(Om / l_m)))) - Float64(Float64(Float64(n * Float64(l_m / Om)) * Float64(l_m / Om)) * Float64(U - U_42_)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Power[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(l$95$m / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 37.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.3

         \[\leadsto {\left(\left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.3

          \[\leadsto {\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

   6. Applied rewrites 37.3%

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

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

   1. Initial program 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 52.8

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

   3. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

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

   1. Initial program 48.7%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 28.2

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

   4. Applied rewrites 28.2%

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

Alternative 3: 60.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

   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 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 52.8

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

   3. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

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

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

Alternative 4: 60.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n + n) * U)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_3 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(t_3 - Float64(Float64(Float64(n * Float64(l_m / Om)) * Float64(l_m / Om)) * Float64(U - U_42_)))));
	else
		tmp = Float64(t_3 * t_1) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(t$95$3 - N[(N[(N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$3 * t$95$1), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

   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 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 52.8

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

   3. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.9

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

   7. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-/.f64 N/A

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

      8. div-flip N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 49.8

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. div-flip N/A

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. lift-/.f64 N/A

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

      24. lower-*.f64 53.9

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

   9. Applied rewrites 53.9%

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

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

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

Alternative 5: 54.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.5 \cdot 10^{-158}:\\ \;\;\;\;{\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;l\_m \leq 3.85 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{l\_m \cdot l\_m}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8.5e-158)
   (pow (* (* (* t n) U) 2.0) 0.5)
   (if (<= l_m 3.85e+62)
     (sqrt
      (*
       (+ n n)
       (*
        (fma
         (- U* U)
         (* (/ (* l_m l_m) (* Om Om)) n)
         (fma -2.0 (/ (* l_m l_m) Om) t))
        U)))
     (pow (* (fma (* (/ l_m Om) l_m) -2.0 t) (* (+ n n) U)) 0.5))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8.5e-158) {
		tmp = pow((((t * n) * U) * 2.0), 0.5);
	} else if (l_m <= 3.85e+62) {
		tmp = sqrt(((n + n) * (fma((U_42_ - U), (((l_m * l_m) / (Om * Om)) * n), fma(-2.0, ((l_m * l_m) / Om), t)) * U)));
	} else {
		tmp = pow((fma(((l_m / Om) * l_m), -2.0, t) * ((n + n) * U)), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8.5e-158)
		tmp = Float64(Float64(Float64(t * n) * U) * 2.0) ^ 0.5;
	elseif (l_m <= 3.85e+62)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(U_42_ - U), Float64(Float64(Float64(l_m * l_m) / Float64(Om * Om)) * n), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)) * U)));
	else
		tmp = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * Float64(Float64(n + n) * U)) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 8.5e-158], N[Power[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 3.85e+62], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.49999999999999944e-158

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 37.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.3

         \[\leadsto {\left(\left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.3

          \[\leadsto {\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

   6. Applied rewrites 37.3%

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

   if 8.49999999999999944e-158 < l < 3.8500000000000001e62

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 48.7

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

   3. Applied rewrites 43.6%

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

   if 3.8500000000000001e62 < l

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

Alternative 6: 53.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 6.8 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(U* - U, \frac{l\_m \cdot l\_m}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.8e-172)
   (pow (* (* (* t n) U) 2.0) 0.5)
   (if (<= l_m 2e+88)
     (sqrt
      (*
       (*
        (fma
         (- U* U)
         (* (/ (* l_m l_m) (* Om Om)) n)
         (fma -2.0 (/ (* l_m l_m) Om) t))
        (+ n n))
       U))
     (pow (* (fma (* (/ l_m Om) l_m) -2.0 t) (* (+ n n) U)) 0.5))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 6.8e-172) {
		tmp = pow((((t * n) * U) * 2.0), 0.5);
	} else if (l_m <= 2e+88) {
		tmp = sqrt(((fma((U_42_ - U), (((l_m * l_m) / (Om * Om)) * n), fma(-2.0, ((l_m * l_m) / Om), t)) * (n + n)) * U));
	} else {
		tmp = pow((fma(((l_m / Om) * l_m), -2.0, t) * ((n + n) * U)), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.8e-172)
		tmp = Float64(Float64(Float64(t * n) * U) * 2.0) ^ 0.5;
	elseif (l_m <= 2e+88)
		tmp = sqrt(Float64(Float64(fma(Float64(U_42_ - U), Float64(Float64(Float64(l_m * l_m) / Float64(Om * Om)) * n), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)) * Float64(n + n)) * U));
	else
		tmp = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * Float64(Float64(n + n) * U)) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 6.8e-172], N[Power[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 2e+88], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.7999999999999997e-172

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 37.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.3

         \[\leadsto {\left(\left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.3

          \[\leadsto {\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

   6. Applied rewrites 37.3%

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

   if 6.7999999999999997e-172 < l < 1.99999999999999992e88

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   if 1.99999999999999992e88 < l

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

Alternative 7: 53.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ \mathbf{if}\;l\_m \leq 3.4 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;l\_m \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(U* - U, \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right) \cdot n, t\_1\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_1 \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l_m Om) l_m) -2.0 t)))
   (if (<= l_m 3.4e-172)
     (pow (* (* (* t n) U) 2.0) 0.5)
     (if (<= l_m 1.1e+81)
       (sqrt
        (* (* (fma (- U* U) (* (* l_m (/ l_m (* Om Om))) n) t_1) (+ n n)) U))
       (pow (* t_1 (* (+ n n) U)) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t);
	double tmp;
	if (l_m <= 3.4e-172) {
		tmp = pow((((t * n) * U) * 2.0), 0.5);
	} else if (l_m <= 1.1e+81) {
		tmp = sqrt(((fma((U_42_ - U), ((l_m * (l_m / (Om * Om))) * n), t_1) * (n + n)) * U));
	} else {
		tmp = pow((t_1 * ((n + n) * U)), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	tmp = 0.0
	if (l_m <= 3.4e-172)
		tmp = Float64(Float64(Float64(t * n) * U) * 2.0) ^ 0.5;
	elseif (l_m <= 1.1e+81)
		tmp = sqrt(Float64(Float64(fma(Float64(U_42_ - U), Float64(Float64(l_m * Float64(l_m / Float64(Om * Om))) * n), t_1) * Float64(n + n)) * U));
	else
		tmp = Float64(t_1 * Float64(Float64(n + n) * U)) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[l$95$m, 3.4e-172], N[Power[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 1.1e+81], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + t$95$1), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$1 * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.3999999999999999e-172

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 37.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.3

         \[\leadsto {\left(\left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.3

          \[\leadsto {\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

   6. Applied rewrites 37.3%

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

   if 3.3999999999999999e-172 < l < 1.09999999999999993e81

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.9

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 44.9

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. div-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 1.09999999999999993e81 < l

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

Alternative 8: 53.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 6.8 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(U*, \frac{l\_m \cdot l\_m}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.8e-172)
   (pow (* (* (* t n) U) 2.0) 0.5)
   (if (<= l_m 2e+88)
     (sqrt
      (*
       (*
        (fma
         U*
         (* (/ (* l_m l_m) (* Om Om)) n)
         (fma -2.0 (/ (* l_m l_m) Om) t))
        (+ n n))
       U))
     (pow (* (fma (* (/ l_m Om) l_m) -2.0 t) (* (+ n n) U)) 0.5))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 6.8e-172) {
		tmp = pow((((t * n) * U) * 2.0), 0.5);
	} else if (l_m <= 2e+88) {
		tmp = sqrt(((fma(U_42_, (((l_m * l_m) / (Om * Om)) * n), fma(-2.0, ((l_m * l_m) / Om), t)) * (n + n)) * U));
	} else {
		tmp = pow((fma(((l_m / Om) * l_m), -2.0, t) * ((n + n) * U)), 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.8e-172)
		tmp = Float64(Float64(Float64(t * n) * U) * 2.0) ^ 0.5;
	elseif (l_m <= 2e+88)
		tmp = sqrt(Float64(Float64(fma(U_42_, Float64(Float64(Float64(l_m * l_m) / Float64(Om * Om)) * n), fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)) * Float64(n + n)) * U));
	else
		tmp = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * Float64(Float64(n + n) * U)) ^ 0.5;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 6.8e-172], N[Power[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l$95$m, 2e+88], N[Sqrt[N[(N[(N[(U$42$ * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.7999999999999997e-172

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 37.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.3

         \[\leadsto {\left(\left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.3

          \[\leadsto {\left(\left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}^{0.5} \]

   6. Applied rewrites 37.3%

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

   if 6.7999999999999997e-172 < l < 1.99999999999999992e88

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in U around 0

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

   6. Applied rewrites 43.6%

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

   if 1.99999999999999992e88 < l

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

Alternative 9: 53.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ t_2 := {\left(t\_1 \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{0.5}\\ \mathbf{if}\;n \leq -6 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot t\_1\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l_m Om) l_m) -2.0 t))
        (t_2 (pow (* t_1 (* (+ n n) U)) 0.5)))
   (if (<= n -6e-33) t_2 (if (<= n 2e-70) (sqrt (* (* (+ n n) t_1) U)) t_2))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t);
	double t_2 = pow((t_1 * ((n + n) * U)), 0.5);
	double tmp;
	if (n <= -6e-33) {
		tmp = t_2;
	} else if (n <= 2e-70) {
		tmp = sqrt((((n + n) * t_1) * U));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	t_2 = Float64(t_1 * Float64(Float64(n + n) * U)) ^ 0.5
	tmp = 0.0
	if (n <= -6e-33)
		tmp = t_2;
	elseif (n <= 2e-70)
		tmp = sqrt(Float64(Float64(Float64(n + n) * t_1) * U));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t$95$1 * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[n, -6e-33], t$95$2, If[LessEqual[n, 2e-70], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.0000000000000003e-33 or 1.99999999999999999e-70 < n

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f64 49.2

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

   8. Applied rewrites 51.7%

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

   if -6.0000000000000003e-33 < n < 1.99999999999999999e-70

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 43.4

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

   8. Applied rewrites 47.5%

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

Alternative 10: 50.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n + n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ \mathbf{if}\;U \leq 4 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{t\_1 \cdot U}\\ \mathbf{elif}\;U \leq 5.8 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (+ n n) (fma (* (/ l_m Om) l_m) -2.0 t))))
   (if (<= U 4e-299)
     (sqrt (* t_1 U))
     (if (<= U 5.8e-238) (sqrt (* (+ n n) (* U t))) (* (sqrt t_1) (sqrt U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n + n) * fma(((l_m / Om) * l_m), -2.0, t);
	double tmp;
	if (U <= 4e-299) {
		tmp = sqrt((t_1 * U));
	} else if (U <= 5.8e-238) {
		tmp = sqrt(((n + n) * (U * t)));
	} else {
		tmp = sqrt(t_1) * sqrt(U);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n + n) * fma(Float64(Float64(l_m / Om) * l_m), -2.0, t))
	tmp = 0.0
	if (U <= 4e-299)
		tmp = sqrt(Float64(t_1 * U));
	elseif (U <= 5.8e-238)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	else
		tmp = Float64(sqrt(t_1) * sqrt(U));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n + n), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, 4e-299], N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 5.8e-238], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if U < 3.99999999999999997e-299

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 43.4

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

   8. Applied rewrites 47.5%

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

   if 3.99999999999999997e-299 < U < 5.7999999999999997e-238

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.9

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

   6. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 34.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.4

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

   10. Applied rewrites 34.4%

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

   if 5.7999999999999997e-238 < U

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 27.8%

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

Alternative 11: 48.5% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l_m Om) l_m) -2.0 t)))
   (if (<= n -10.0)
     (sqrt (* t_1 (* (+ n n) U)))
     (sqrt (* (* (+ n n) t_1) U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t);
	double tmp;
	if (n <= -10.0) {
		tmp = sqrt((t_1 * ((n + n) * U)));
	} else {
		tmp = sqrt((((n + n) * t_1) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	tmp = 0.0
	if (n <= -10.0)
		tmp = sqrt(Float64(t_1 * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(n + n) * t_1) * U));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[n, -10.0], N[Sqrt[N[(t$95$1 * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -10

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 42.5

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

   8. Applied rewrites 46.3%

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

   if -10 < n

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 43.4

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

   8. Applied rewrites 47.5%

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

Alternative 12: 48.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.9

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

   6. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 34.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.4

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

   10. Applied rewrites 34.4%

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

   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*))))

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.4

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

   6. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 42.5

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

   8. Applied rewrites 46.3%

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

Alternative 13: 37.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.9

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

   6. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 34.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.4

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

   10. Applied rewrites 34.4%

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

   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*))))

   1. Initial program 48.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 35.2

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

   4. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.9

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

   6. Applied rewrites 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 34.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

Alternative 14: 34.9% accurate, 4.7× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (+ n n) U) t)))

C

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

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.7%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 35.2

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

4. Applied rewrites 35.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 34.9

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

6. Applied rewrites 34.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. count-2-rev N/A

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

   8. lift-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 34.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 34.9

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

8. Applied rewrites 34.9%

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

Reproduce

herbie shell --seed 2025150 
(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*))))))