Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 64.4%

Time: 9.6s

Alternatives: 15

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 64.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t\_1 + \left(-\left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}, -2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l_m (/ l_m Om)) t))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* t_1 n) U) 2.0))
     (if (<= t_3 1e+307)
       (sqrt (* t_2 (+ t_1 (* (- (* (* (/ l_m Om) (/ l_m Om)) n)) (- U U*)))))
       (*
        (sqrt
         (*
          U
          (* n (fma -1.0 (/ (* n (- U U*)) (* Om Om)) (* -2.0 (/ 1.0 Om))))))
        (* l_m (sqrt 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 = fma(-2.0, (l_m * (l_m / Om)), t);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else if (t_3 <= 1e+307) {
		tmp = sqrt((t_2 * (t_1 + (-(((l_m / Om) * (l_m / Om)) * n) * (U - U_42_)))));
	} else {
		tmp = sqrt((U * (n * fma(-1.0, ((n * (U - U_42_)) / (Om * Om)), (-2.0 * (1.0 / Om)))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= 1e+307)
		tmp = sqrt(Float64(t_2 * Float64(t_1 + Float64(Float64(-Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n)) * Float64(U - U_42_)))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * fma(-1.0, Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om)), Float64(-2.0 * Float64(1.0 / Om)))))) * Float64(l_m * sqrt(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[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$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$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+307], N[Sqrt[N[(t$95$2 * N[(t$95$1 + N[((-N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(-1.0 * N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $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 10.2%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 96.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 96.8%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 22.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 29.8%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   6. Applied rewrites 45.1%

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

Alternative 2: 60.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(-\frac{\mathsf{fma}\left(l\_m \cdot l\_m, \frac{\left(U - U*\right) \cdot n}{Om}, \left(l\_m \cdot l\_m\right) \cdot 2\right)}{Om}\right) + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}, -2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

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
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
     (if (<= t_2 1e+307)
       (sqrt
        (*
         t_1
         (+
          (-
           (/ (fma (* l_m l_m) (/ (* (- U U*) n) Om) (* (* l_m l_m) 2.0)) Om))
          t)))
       (*
        (sqrt
         (*
          U
          (* n (fma -1.0 (/ (* n (- U U*)) (* Om Om)) (* -2.0 (/ 1.0 Om))))))
        (* l_m (sqrt 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;
	double t_2 = t_1 * ((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((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * (-(fma((l_m * l_m), (((U - U_42_) * n) / Om), ((l_m * l_m) * 2.0)) / Om) + t)));
	} else {
		tmp = sqrt((U * (n * fma(-1.0, ((n * (U - U_42_)) / (Om * Om)), (-2.0 * (1.0 / Om)))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_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(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * Float64(Float64(-Float64(fma(Float64(l_m * l_m), Float64(Float64(Float64(U - U_42_) * n) / Om), Float64(Float64(l_m * l_m) * 2.0)) / Om)) + t)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * fma(-1.0, Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om)), Float64(-2.0 * Float64(1.0 / Om)))))) * Float64(l_m * sqrt(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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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$2, 0.0], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[((-N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(-1.0 * N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $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 10.2%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 96.8%

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

   2. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 85.2%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 22.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 29.8%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   6. Applied rewrites 45.1%

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

Alternative 3: 59.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}, -2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l_m (/ l_m Om)) t))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (sqrt (* (* (* t_1 n) U) 2.0))
     (if (<= t_3 1e+307)
       (sqrt (* t_2 t_1))
       (*
        (sqrt
         (*
          U
          (* n (fma -1.0 (/ (* n (- U U*)) (* Om Om)) (* -2.0 (/ 1.0 Om))))))
        (* l_m (sqrt 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 = fma(-2.0, (l_m * (l_m / Om)), t);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else if (t_3 <= 1e+307) {
		tmp = sqrt((t_2 * t_1));
	} else {
		tmp = sqrt((U * (n * fma(-1.0, ((n * (U - U_42_)) / (Om * Om)), (-2.0 * (1.0 / Om)))))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= 1e+307)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * fma(-1.0, Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om)), Float64(-2.0 * Float64(1.0 / Om)))))) * Float64(l_m * sqrt(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[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$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$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+307], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(-1.0 * N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $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 10.2%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 96.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 22.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 29.8%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   6. Applied rewrites 45.1%

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

Alternative 4: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(-\left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;n \leq -8 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 8.7 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ t (* (- (* (* (/ l_m Om) (/ l_m Om)) n)) (- U U*)))))))
   (if (<= n -8e+37)
     t_1
     (if (<= n 8.7e-162)
       (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * (t + (-(((l_m / Om) * (l_m / Om)) * n) * (U - U_42_)))));
	double tmp;
	if (n <= -8e+37) {
		tmp = t_1;
	} else if (n <= 8.7e-162) {
		tmp = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(-Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (n <= -8e+37)
		tmp = t_1;
	elseif (n <= 8.7e-162)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	else
		tmp = t_1;
	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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[((-N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -8e+37], t$95$1, If[LessEqual[n, 8.7e-162], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -7.99999999999999963e37 or 8.7000000000000003e-162 < n

   1. Initial program 54.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

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

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

      12. pow2 N/A

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

      13. lower-+.f64 N/A

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

   3. Applied rewrites 58.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 58.5%

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

   if -7.99999999999999963e37 < n < 8.7000000000000003e-162

   1. Initial program 44.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 53.1

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

   4. Applied rewrites 53.1%

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

Alternative 5: 54.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \frac{l\_m \cdot l\_m}{Om \cdot Om}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;n \leq -5.8 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- t (* (* n (/ (* l_m l_m) (* Om Om))) (- U U*)))))))
   (if (<= n -5.8e+60)
     t_1
     (if (<= n 4e-71)
       (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * (t - ((n * ((l_m * l_m) / (Om * Om))) * (U - U_42_)))));
	double tmp;
	if (n <= -5.8e+60) {
		tmp = t_1;
	} else if (n <= 4e-71) {
		tmp = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l_m * l_m) / Float64(Om * Om))) * Float64(U - U_42_)))))
	tmp = 0.0
	if (n <= -5.8e+60)
		tmp = t_1;
	elseif (n <= 4e-71)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	else
		tmp = t_1;
	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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -5.8e+60], t$95$1, If[LessEqual[n, 4e-71], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -5.79999999999999999e60 or 3.9999999999999997e-71 < n

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 60.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 53.5

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

   6. Applied rewrites 53.5%

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

   if -5.79999999999999999e60 < n < 3.9999999999999997e-71

   1. Initial program 45.8%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 6: 54.3% accurate, 0.4× speedup

Math

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

FPCore

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

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(Float64(Float64(l_m * l_m) * n) / Om) * Float64(U_42_ / Om))));
	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[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$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$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $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 10.2%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

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

   1. Initial program 68.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 28.4

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

   4. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 30.2

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

   6. Applied rewrites 30.2%

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

Alternative 7: 53.6% accurate, 0.4× speedup

Math

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

FPCore

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

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(Float64(Float64(l_m * l_m) * n) * U_42_) / Float64(Om * Om))));
	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[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$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$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U$42$), $MachinePrecision] / N[(Om * Om), $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 10.2%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

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

   1. Initial program 68.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 28.4

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

   4. Applied rewrites 28.4%

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

Alternative 8: 53.6% accurate, 0.4× speedup

Math

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

FPCore

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

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l_m) / Om));
	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[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$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$3, 0.0], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.2%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

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

   1. Initial program 68.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 23.5

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

   4. Applied rewrites 23.5%

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

Alternative 9: 49.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 2.4e+160)
   (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
   (* (sqrt (* (* 2.0 n) U)) (sqrt 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 (t <= 2.4e+160) {
		tmp = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 2.4e+160)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 2.4e+160], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000001e160

   1. Initial program 50.9%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 48.6

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

   4. Applied rewrites 48.6%

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

   if 2.4000000000000001e160 < t

   1. Initial program 45.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 42.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 55.5

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

   6. Applied rewrites 55.5%

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

Alternative 10: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -7.9 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\left(\left(\left(-2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -7.9e-159)
   (sqrt (* (* (* t n) U) 2.0))
   (if (<= t 1.7e-106)
     (sqrt (* (* (* (* -2.0 (/ (* l_m l_m) Om)) n) U) 2.0))
     (* (sqrt (* (* 2.0 n) U)) (sqrt 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 (t <= -7.9e-159) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else if (t <= 1.7e-106) {
		tmp = sqrt(((((-2.0 * ((l_m * l_m) / Om)) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(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 (t <= (-7.9d-159)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else if (t <= 1.7d-106) then
        tmp = sqrt((((((-2.0d0) * ((l_m * l_m) / om)) * n) * u) * 2.0d0))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(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 (t <= -7.9e-159) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else if (t <= 1.7e-106) {
		tmp = Math.sqrt(((((-2.0 * ((l_m * l_m) / Om)) * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -7.9e-159:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	elif t <= 1.7e-106:
		tmp = math.sqrt(((((-2.0 * ((l_m * l_m) / Om)) * n) * U) * 2.0))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -7.9e-159)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	elseif (t <= 1.7e-106)
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(l_m * l_m) / Om)) * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(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 (t <= -7.9e-159)
		tmp = sqrt((((t * n) * U) * 2.0));
	elseif (t <= 1.7e-106)
		tmp = sqrt(((((-2.0 * ((l_m * l_m) / Om)) * n) * U) * 2.0));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(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[t, -7.9e-159], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.7e-106], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.89999999999999972e-159

   1. Initial program 50.9%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.9

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

   4. Applied rewrites 40.9%

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

   if -7.89999999999999972e-159 < t < 1.69999999999999991e-106

   1. Initial program 47.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 40.8

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

   4. Applied rewrites 40.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 22.2

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

   7. Applied rewrites 22.2%

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

   if 1.69999999999999991e-106 < t

   1. Initial program 51.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 47.5

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

   6. Applied rewrites 47.5%

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

Alternative 11: 39.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_1 \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot l\_m}{Om}\\ \end{array} \end{array} \]

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 n) U) 2.0))
     (if (<= t_1 1e+307)
       (sqrt (* (* 2.0 (* U n)) t))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l_m) Om))))))

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 * n) * U) * 2.0));
	} else if (t_1 <= 1e+307) {
		tmp = sqrt(((2.0 * (U * n)) * t));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l_m) / Om);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else if (t_1 <= 1d+307) then
        tmp = sqrt(((2.0d0 * (u * n)) * t))
    else
        tmp = sqrt((u_42 * u)) * (((sqrt(2.0d0) * n) * l_m) / om)
    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 t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else if (t_1 <= 1e+307) {
		tmp = Math.sqrt(((2.0 * (U * n)) * t));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l_m) / Om);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	elif t_1 <= 1e+307:
		tmp = math.sqrt(((2.0 * (U * n)) * t))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l_m) / Om)
	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 * n) * U) * 2.0));
	elseif (t_1 <= 1e+307)
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * n)) * t));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l_m) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((((t * n) * U) * 2.0));
	elseif (t_1 <= 1e+307)
		tmp = sqrt(((2.0 * (U * n)) * t));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l_m) / Om);
	end
	tmp_2 = 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), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+307], N[Sqrt[N[(N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.2%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 96.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 85.8

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

   10. Applied rewrites 85.8%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 74.2%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 22.5%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 21.6

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

   4. Applied rewrites 21.6%

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

Alternative 12: 38.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -7.8e-159)
   (sqrt (* (* (* t n) U) 2.0))
   (if (<= t 1.65e-106)
     (sqrt (* -4.0 (/ (* U (* (* l_m l_m) n)) Om)))
     (* (sqrt (* (* 2.0 n) U)) (sqrt 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 (t <= -7.8e-159) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else if (t <= 1.65e-106) {
		tmp = sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(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 (t <= (-7.8d-159)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else if (t <= 1.65d-106) then
        tmp = sqrt(((-4.0d0) * ((u * ((l_m * l_m) * n)) / om)))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(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 (t <= -7.8e-159) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else if (t <= 1.65e-106) {
		tmp = Math.sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -7.8e-159:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	elif t <= 1.65e-106:
		tmp = math.sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -7.8e-159)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	elseif (t <= 1.65e-106)
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(Float64(l_m * l_m) * n)) / Om)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(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 (t <= -7.8e-159)
		tmp = sqrt((((t * n) * U) * 2.0));
	elseif (t <= 1.65e-106)
		tmp = sqrt((-4.0 * ((U * ((l_m * l_m) * n)) / Om)));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(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[t, -7.8e-159], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.65e-106], N[Sqrt[N[(-4.0 * N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.79999999999999953e-159

   1. Initial program 50.9%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.9

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

   4. Applied rewrites 40.9%

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

   if -7.79999999999999953e-159 < t < 1.65000000000000008e-106

   1. Initial program 47.0%

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

   2. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 36.2

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.4

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

   7. Applied rewrites 21.4%

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

   if 1.65000000000000008e-106 < t

   1. Initial program 51.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 47.5

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

   6. Applied rewrites 47.5%

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

Alternative 13: 38.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -4e-310)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* (* 2.0 n) U)) (sqrt 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 (t <= -4e-310) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(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 (t <= (-4d-310)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(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 (t <= -4e-310) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -4e-310:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -4e-310)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -4e-310)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(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[t, -4e-310], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.999999999999988e-310

   1. Initial program 49.9%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.6

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

   4. Applied rewrites 35.6%

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

   if -3.999999999999988e-310 < t

   1. Initial program 50.4%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 35.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 42.5

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

   6. Applied rewrites 42.5%

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

Alternative 14: 37.9% accurate, 0.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (* 2.0 (* U n)) 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 (sqrt((((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((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(((2.0 * (U * n)) * 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 (sqrt((((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((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((2.0d0 * (u * n)) * 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 (Math.sqrt((((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((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * (U * n)) * t));
	}
	return tmp;
}

Python

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (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_))))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * n)) * 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 (sqrt((((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((((t * n) * U) * 2.0));
	else
		tmp = sqrt(((2.0 * (U * n)) * 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[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], 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 11.6%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.6

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

   4. Applied rewrites 32.6%

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

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

   1. Initial program 55.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 56.3

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

   7. Applied rewrites 56.3%

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

   8. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 39.2%

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

Alternative 15: 36.1% accurate, 4.6× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* t n) U) 2.0)))

C

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

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((((t * n) * u) * 2.0d0))
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((((t * n) * U) * 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.2%

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

2. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 36.1

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

4. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2025123 
(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*))))))