Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 90.6%

Time: 4.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

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(c0, a, v, l)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

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(c0, a, v, l)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\left(A\_m \cdot c0\right) \cdot \sqrt{\frac{V\_m}{A\_m \cdot l\_m}}}{V\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{l\_m}}{V\_m} \cdot A\_m}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{V\_m \cdot \frac{A\_m}{l\_m}}}{V\_m}\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m)))))
   (if (<= t_0 2e-157)
     (/ (* (* A_m c0) (sqrt (/ V_m (* A_m l_m)))) V_m)
     (if (<= t_0 1e+103)
       (* c0 (sqrt (* (/ (/ 1.0 l_m) V_m) A_m)))
       (* c0 (/ (sqrt (* V_m (/ A_m l_m))) V_m))))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = ((A_m * c0) * sqrt((V_m / (A_m * l_m)))) / V_m;
	} else if (t_0 <= 1e+103) {
		tmp = c0 * sqrt((((1.0 / l_m) / V_m) * A_m));
	} else {
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    if (t_0 <= 2d-157) then
        tmp = ((a_m * c0) * sqrt((v_m / (a_m * l_m)))) / v_m
    else if (t_0 <= 1d+103) then
        tmp = c0 * sqrt((((1.0d0 / l_m) / v_m) * a_m))
    else
        tmp = c0 * (sqrt((v_m * (a_m / l_m))) / v_m)
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = ((A_m * c0) * Math.sqrt((V_m / (A_m * l_m)))) / V_m;
	} else if (t_0 <= 1e+103) {
		tmp = c0 * Math.sqrt((((1.0 / l_m) / V_m) * A_m));
	} else {
		tmp = c0 * (Math.sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	tmp = 0
	if t_0 <= 2e-157:
		tmp = ((A_m * c0) * math.sqrt((V_m / (A_m * l_m)))) / V_m
	elif t_0 <= 1e+103:
		tmp = c0 * math.sqrt((((1.0 / l_m) / V_m) * A_m))
	else:
		tmp = c0 * (math.sqrt((V_m * (A_m / l_m))) / V_m)
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	tmp = 0.0
	if (t_0 <= 2e-157)
		tmp = Float64(Float64(Float64(A_m * c0) * sqrt(Float64(V_m / Float64(A_m * l_m)))) / V_m);
	elseif (t_0 <= 1e+103)
		tmp = Float64(c0 * sqrt(Float64(Float64(Float64(1.0 / l_m) / V_m) * A_m)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(V_m * Float64(A_m / l_m))) / V_m));
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	tmp = 0.0;
	if (t_0 <= 2e-157)
		tmp = ((A_m * c0) * sqrt((V_m / (A_m * l_m)))) / V_m;
	elseif (t_0 <= 1e+103)
		tmp = c0 * sqrt((((1.0 / l_m) / V_m) * A_m));
	else
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-157], N[(N[(N[(A$95$m * c0), $MachinePrecision] * N[Sqrt[N[(V$95$m / N[(A$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / V$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e+103], N[(c0 * N[Sqrt[N[(N[(N[(1.0 / l$95$m), $MachinePrecision] / V$95$m), $MachinePrecision] * A$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(V$95$m * N[(A$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / V$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.99999999999999989e-157

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\sqrt{V \cdot \frac{A}{\ell}} \cdot \frac{c0}{V}} \]

   7. Taylor expanded in A around inf

      \[\leadsto \frac{A \cdot \left(c0 \cdot \sqrt{\frac{V}{A \cdot \ell}}\right)}{\color{blue}{V}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{A \cdot \left(c0 \cdot \sqrt{\frac{V}{A \cdot \ell}}\right)}{V} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{V} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{V} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{V} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{V} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{\ell \cdot A}}}{V} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{\ell \cdot A}}}{V} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{V} \]

      9. lower-*.f64 55.9

         \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{V} \]

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\left(A \cdot c0\right) \cdot \sqrt{\frac{V}{A \cdot \ell}}}{\color{blue}{V}} \]

   if 1.99999999999999989e-157 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 1e103

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. mult-flip N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      7. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      8. lower-*.f64 72.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

   3. Applied rewrites 72.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V} \cdot A}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V}} \cdot A} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{V}} \cdot A} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{V}} \cdot A} \]

      5. lower-/.f64 73.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\ell}}}{V} \cdot A} \]

   5. Applied rewrites 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{V}} \cdot A} \]

   if 1e103 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      2. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      4. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      5. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      6. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      7. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      8. mult-flip-rev N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      10. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      11. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      12. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      13. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      14. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot \color{blue}{V}}{\ell}}}{V} \]

      15. mult-flip N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      16. associate-/l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      17. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      18. lower-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{\color{blue}{V}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{V} \]

      20. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{V \cdot A}{\ell}}}{V} \]

      21. associate-/l* N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      22. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      23. lower-*.f64 69.8

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

   6. Applied rewrites 69.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{V \cdot \frac{A}{\ell}}}{V}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{\frac{A\_m}{V\_m} \cdot l\_m} \cdot c0}{l\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{l\_m}}{V\_m} \cdot A\_m}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{V\_m \cdot \frac{A\_m}{l\_m}}}{V\_m}\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m)))))
   (if (<= t_0 2e-157)
     (/ (* (sqrt (* (/ A_m V_m) l_m)) c0) l_m)
     (if (<= t_0 1e+103)
       (* c0 (sqrt (* (/ (/ 1.0 l_m) V_m) A_m)))
       (* c0 (/ (sqrt (* V_m (/ A_m l_m))) V_m))))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 1e+103) {
		tmp = c0 * sqrt((((1.0 / l_m) / V_m) * A_m));
	} else {
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    if (t_0 <= 2d-157) then
        tmp = (sqrt(((a_m / v_m) * l_m)) * c0) / l_m
    else if (t_0 <= 1d+103) then
        tmp = c0 * sqrt((((1.0d0 / l_m) / v_m) * a_m))
    else
        tmp = c0 * (sqrt((v_m * (a_m / l_m))) / v_m)
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (Math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 1e+103) {
		tmp = c0 * Math.sqrt((((1.0 / l_m) / V_m) * A_m));
	} else {
		tmp = c0 * (Math.sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	tmp = 0
	if t_0 <= 2e-157:
		tmp = (math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m
	elif t_0 <= 1e+103:
		tmp = c0 * math.sqrt((((1.0 / l_m) / V_m) * A_m))
	else:
		tmp = c0 * (math.sqrt((V_m * (A_m / l_m))) / V_m)
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	tmp = 0.0
	if (t_0 <= 2e-157)
		tmp = Float64(Float64(sqrt(Float64(Float64(A_m / V_m) * l_m)) * c0) / l_m);
	elseif (t_0 <= 1e+103)
		tmp = Float64(c0 * sqrt(Float64(Float64(Float64(1.0 / l_m) / V_m) * A_m)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(V_m * Float64(A_m / l_m))) / V_m));
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	tmp = 0.0;
	if (t_0 <= 2e-157)
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	elseif (t_0 <= 1e+103)
		tmp = c0 * sqrt((((1.0 / l_m) / V_m) * A_m));
	else
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-157], N[(N[(N[Sqrt[N[(N[(A$95$m / V$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e+103], N[(c0 * N[Sqrt[N[(N[(N[(1.0 / l$95$m), $MachinePrecision] / V$95$m), $MachinePrecision] * A$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(V$95$m * N[(A$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / V$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.99999999999999989e-157

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]

      2. mult-flip N/A

         \[\leadsto c0 \cdot \left(\sqrt{\frac{A \cdot \ell}{V}} \cdot \color{blue}{\frac{1}{\ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      11. lower-*.f64 62.2

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \color{blue}{\sqrt{\frac{\ell \cdot A}{V}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{\ell \cdot A}{V}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{\ell \cdot A}{V}}}{\color{blue}{\ell}} \]

      7. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(\ell \cdot A\right) \cdot \frac{1}{V}}}{\ell} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(A \cdot \ell\right) \cdot \frac{1}{V}}}{\ell} \]

      9. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\color{blue}{\ell}} \]

   6. Applied rewrites 64.0%

      \[\leadsto \frac{\sqrt{\frac{A}{V} \cdot \ell} \cdot c0}{\color{blue}{\ell}} \]

   if 1.99999999999999989e-157 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 1e103

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. mult-flip N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      7. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      8. lower-*.f64 72.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

   3. Applied rewrites 72.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V} \cdot A}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V}} \cdot A} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{V}} \cdot A} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{V}} \cdot A} \]

      5. lower-/.f64 73.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\ell}}}{V} \cdot A} \]

   5. Applied rewrites 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{V}} \cdot A} \]

   if 1e103 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      2. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      4. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      5. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      6. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      7. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      8. mult-flip-rev N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      10. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      11. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      12. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      13. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      14. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot \color{blue}{V}}{\ell}}}{V} \]

      15. mult-flip N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      16. associate-/l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      17. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      18. lower-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{\color{blue}{V}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{V} \]

      20. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{V \cdot A}{\ell}}}{V} \]

      21. associate-/l* N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      22. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      23. lower-*.f64 69.8

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

   6. Applied rewrites 69.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{V \cdot \frac{A}{\ell}}}{V}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{\frac{A\_m}{V\_m} \cdot l\_m} \cdot c0}{l\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+117}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{V\_m}}{l\_m} \cdot A\_m}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{V\_m \cdot \frac{A\_m}{l\_m}}}{V\_m}\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m)))))
   (if (<= t_0 2e-157)
     (/ (* (sqrt (* (/ A_m V_m) l_m)) c0) l_m)
     (if (<= t_0 5e+117)
       (* c0 (sqrt (* (/ (/ 1.0 V_m) l_m) A_m)))
       (* c0 (/ (sqrt (* V_m (/ A_m l_m))) V_m))))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 5e+117) {
		tmp = c0 * sqrt((((1.0 / V_m) / l_m) * A_m));
	} else {
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    if (t_0 <= 2d-157) then
        tmp = (sqrt(((a_m / v_m) * l_m)) * c0) / l_m
    else if (t_0 <= 5d+117) then
        tmp = c0 * sqrt((((1.0d0 / v_m) / l_m) * a_m))
    else
        tmp = c0 * (sqrt((v_m * (a_m / l_m))) / v_m)
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (Math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 5e+117) {
		tmp = c0 * Math.sqrt((((1.0 / V_m) / l_m) * A_m));
	} else {
		tmp = c0 * (Math.sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	tmp = 0
	if t_0 <= 2e-157:
		tmp = (math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m
	elif t_0 <= 5e+117:
		tmp = c0 * math.sqrt((((1.0 / V_m) / l_m) * A_m))
	else:
		tmp = c0 * (math.sqrt((V_m * (A_m / l_m))) / V_m)
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	tmp = 0.0
	if (t_0 <= 2e-157)
		tmp = Float64(Float64(sqrt(Float64(Float64(A_m / V_m) * l_m)) * c0) / l_m);
	elseif (t_0 <= 5e+117)
		tmp = Float64(c0 * sqrt(Float64(Float64(Float64(1.0 / V_m) / l_m) * A_m)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(V_m * Float64(A_m / l_m))) / V_m));
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	tmp = 0.0;
	if (t_0 <= 2e-157)
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	elseif (t_0 <= 5e+117)
		tmp = c0 * sqrt((((1.0 / V_m) / l_m) * A_m));
	else
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-157], N[(N[(N[Sqrt[N[(N[(A$95$m / V$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+117], N[(c0 * N[Sqrt[N[(N[(N[(1.0 / V$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * A$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(V$95$m * N[(A$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / V$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.99999999999999989e-157

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]

      2. mult-flip N/A

         \[\leadsto c0 \cdot \left(\sqrt{\frac{A \cdot \ell}{V}} \cdot \color{blue}{\frac{1}{\ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      11. lower-*.f64 62.2

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \color{blue}{\sqrt{\frac{\ell \cdot A}{V}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{\ell \cdot A}{V}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{\ell \cdot A}{V}}}{\color{blue}{\ell}} \]

      7. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(\ell \cdot A\right) \cdot \frac{1}{V}}}{\ell} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(A \cdot \ell\right) \cdot \frac{1}{V}}}{\ell} \]

      9. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\color{blue}{\ell}} \]

   6. Applied rewrites 64.0%

      \[\leadsto \frac{\sqrt{\frac{A}{V} \cdot \ell} \cdot c0}{\color{blue}{\ell}} \]

   if 1.99999999999999989e-157 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 4.99999999999999983e117

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. mult-flip N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      7. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      8. lower-*.f64 72.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

   3. Applied rewrites 72.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V} \cdot A}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      2. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell}} \cdot A} \]

      3. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      4. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell}} \cdot A} \]

      5. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell}} \cdot A} \]

      6. lift-/.f64 73.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{V}}}{\ell} \cdot A} \]

   5. Applied rewrites 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\ell}} \cdot A} \]

   if 4.99999999999999983e117 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      2. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      4. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      5. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      6. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      7. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      8. mult-flip-rev N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      10. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      11. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      12. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      13. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      14. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot \color{blue}{V}}{\ell}}}{V} \]

      15. mult-flip N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      16. associate-/l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      17. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      18. lower-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{\color{blue}{V}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{V} \]

      20. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{V \cdot A}{\ell}}}{V} \]

      21. associate-/l* N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      22. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      23. lower-*.f64 69.8

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

   6. Applied rewrites 69.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{V \cdot \frac{A}{\ell}}}{V}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{\frac{A\_m}{V\_m} \cdot l\_m} \cdot c0}{l\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+130}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{l\_m \cdot V\_m} \cdot A\_m}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{V\_m \cdot \frac{A\_m}{l\_m}}}{V\_m}\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m)))))
   (if (<= t_0 2e-157)
     (/ (* (sqrt (* (/ A_m V_m) l_m)) c0) l_m)
     (if (<= t_0 2e+130)
       (* c0 (sqrt (* (/ 1.0 (* l_m V_m)) A_m)))
       (* c0 (/ (sqrt (* V_m (/ A_m l_m))) V_m))))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 2e+130) {
		tmp = c0 * sqrt(((1.0 / (l_m * V_m)) * A_m));
	} else {
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    if (t_0 <= 2d-157) then
        tmp = (sqrt(((a_m / v_m) * l_m)) * c0) / l_m
    else if (t_0 <= 2d+130) then
        tmp = c0 * sqrt(((1.0d0 / (l_m * v_m)) * a_m))
    else
        tmp = c0 * (sqrt((v_m * (a_m / l_m))) / v_m)
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (Math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 2e+130) {
		tmp = c0 * Math.sqrt(((1.0 / (l_m * V_m)) * A_m));
	} else {
		tmp = c0 * (Math.sqrt((V_m * (A_m / l_m))) / V_m);
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	tmp = 0
	if t_0 <= 2e-157:
		tmp = (math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m
	elif t_0 <= 2e+130:
		tmp = c0 * math.sqrt(((1.0 / (l_m * V_m)) * A_m))
	else:
		tmp = c0 * (math.sqrt((V_m * (A_m / l_m))) / V_m)
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	tmp = 0.0
	if (t_0 <= 2e-157)
		tmp = Float64(Float64(sqrt(Float64(Float64(A_m / V_m) * l_m)) * c0) / l_m);
	elseif (t_0 <= 2e+130)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / Float64(l_m * V_m)) * A_m)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(V_m * Float64(A_m / l_m))) / V_m));
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	tmp = 0.0;
	if (t_0 <= 2e-157)
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	elseif (t_0 <= 2e+130)
		tmp = c0 * sqrt(((1.0 / (l_m * V_m)) * A_m));
	else
		tmp = c0 * (sqrt((V_m * (A_m / l_m))) / V_m);
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-157], N[(N[(N[Sqrt[N[(N[(A$95$m / V$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e+130], N[(c0 * N[Sqrt[N[(N[(1.0 / N[(l$95$m * V$95$m), $MachinePrecision]), $MachinePrecision] * A$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(V$95$m * N[(A$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / V$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.99999999999999989e-157

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]

      2. mult-flip N/A

         \[\leadsto c0 \cdot \left(\sqrt{\frac{A \cdot \ell}{V}} \cdot \color{blue}{\frac{1}{\ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      11. lower-*.f64 62.2

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \color{blue}{\sqrt{\frac{\ell \cdot A}{V}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{\ell \cdot A}{V}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{\ell \cdot A}{V}}}{\color{blue}{\ell}} \]

      7. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(\ell \cdot A\right) \cdot \frac{1}{V}}}{\ell} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(A \cdot \ell\right) \cdot \frac{1}{V}}}{\ell} \]

      9. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\color{blue}{\ell}} \]

   6. Applied rewrites 64.0%

      \[\leadsto \frac{\sqrt{\frac{A}{V} \cdot \ell} \cdot c0}{\color{blue}{\ell}} \]

   if 1.99999999999999989e-157 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 2.0000000000000001e130

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. mult-flip N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      7. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      8. lower-*.f64 72.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

   3. Applied rewrites 72.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V} \cdot A}} \]

   if 2.0000000000000001e130 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      2. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      4. associate-/l/ N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      5. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      6. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      7. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      8. mult-flip-rev N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\color{blue}{\ell}}}}{V} \]

      10. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      11. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      12. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      13. associate-*l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      14. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot \color{blue}{V}}{\ell}}}{V} \]

      15. mult-flip N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{A \cdot V}}{\ell}}}{V} \]

      16. associate-/l/ N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A \cdot V}{\ell}}}}{V} \]

      17. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A \cdot V}{\ell}}}}{V} \]

      18. lower-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{\color{blue}{V}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A \cdot V}{\ell}}}{V} \]

      20. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{V \cdot A}{\ell}}}{V} \]

      21. associate-/l* N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      22. lift-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

      23. lower-*.f64 69.8

          \[\leadsto c0 \cdot \frac{\sqrt{V \cdot \frac{A}{\ell}}}{V} \]

   6. Applied rewrites 69.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{V \cdot \frac{A}{\ell}}}{V}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{\frac{A\_m}{V\_m} \cdot l\_m} \cdot c0}{l\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+130}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{l\_m \cdot V\_m} \cdot A\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{V\_m \cdot \frac{A\_m}{l\_m}} \cdot \frac{c0}{V\_m}\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m)))))
   (if (<= t_0 2e-157)
     (/ (* (sqrt (* (/ A_m V_m) l_m)) c0) l_m)
     (if (<= t_0 2e+130)
       (* c0 (sqrt (* (/ 1.0 (* l_m V_m)) A_m)))
       (* (sqrt (* V_m (/ A_m l_m))) (/ c0 V_m))))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 2e+130) {
		tmp = c0 * sqrt(((1.0 / (l_m * V_m)) * A_m));
	} else {
		tmp = sqrt((V_m * (A_m / l_m))) * (c0 / V_m);
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    if (t_0 <= 2d-157) then
        tmp = (sqrt(((a_m / v_m) * l_m)) * c0) / l_m
    else if (t_0 <= 2d+130) then
        tmp = c0 * sqrt(((1.0d0 / (l_m * v_m)) * a_m))
    else
        tmp = sqrt((v_m * (a_m / l_m))) * (c0 / v_m)
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = (Math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	} else if (t_0 <= 2e+130) {
		tmp = c0 * Math.sqrt(((1.0 / (l_m * V_m)) * A_m));
	} else {
		tmp = Math.sqrt((V_m * (A_m / l_m))) * (c0 / V_m);
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	tmp = 0
	if t_0 <= 2e-157:
		tmp = (math.sqrt(((A_m / V_m) * l_m)) * c0) / l_m
	elif t_0 <= 2e+130:
		tmp = c0 * math.sqrt(((1.0 / (l_m * V_m)) * A_m))
	else:
		tmp = math.sqrt((V_m * (A_m / l_m))) * (c0 / V_m)
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	tmp = 0.0
	if (t_0 <= 2e-157)
		tmp = Float64(Float64(sqrt(Float64(Float64(A_m / V_m) * l_m)) * c0) / l_m);
	elseif (t_0 <= 2e+130)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / Float64(l_m * V_m)) * A_m)));
	else
		tmp = Float64(sqrt(Float64(V_m * Float64(A_m / l_m))) * Float64(c0 / V_m));
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	tmp = 0.0;
	if (t_0 <= 2e-157)
		tmp = (sqrt(((A_m / V_m) * l_m)) * c0) / l_m;
	elseif (t_0 <= 2e+130)
		tmp = c0 * sqrt(((1.0 / (l_m * V_m)) * A_m));
	else
		tmp = sqrt((V_m * (A_m / l_m))) * (c0 / V_m);
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e-157], N[(N[(N[Sqrt[N[(N[(A$95$m / V$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e+130], N[(c0 * N[Sqrt[N[(N[(1.0 / N[(l$95$m * V$95$m), $MachinePrecision]), $MachinePrecision] * A$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(V$95$m * N[(A$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(c0 / V$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.99999999999999989e-157

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A \cdot \ell}{V}}}{\ell}} \]

      2. mult-flip N/A

         \[\leadsto c0 \cdot \left(\sqrt{\frac{A \cdot \ell}{V}} \cdot \color{blue}{\frac{1}{\ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{1}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{A \cdot \ell}{V}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{A \cdot \ell}{V}}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{A \cdot \ell}{V}} \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      11. lower-*.f64 62.2

          \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \color{blue}{\sqrt{\frac{\ell \cdot A}{V}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\color{blue}{\frac{\ell \cdot A}{V}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\ell} \cdot \sqrt{\frac{\ell \cdot A}{V}} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{\ell \cdot A}{V}}}{\color{blue}{\ell}} \]

      7. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(\ell \cdot A\right) \cdot \frac{1}{V}}}{\ell} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\left(A \cdot \ell\right) \cdot \frac{1}{V}}}{\ell} \]

      9. mult-flip N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\ell} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0 \cdot \sqrt{\frac{A \cdot \ell}{V}}}{\color{blue}{\ell}} \]

   6. Applied rewrites 64.0%

      \[\leadsto \frac{\sqrt{\frac{A}{V} \cdot \ell} \cdot c0}{\color{blue}{\ell}} \]

   if 1.99999999999999989e-157 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 2.0000000000000001e130

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. mult-flip N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      7. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      8. lower-*.f64 72.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

   3. Applied rewrites 72.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V} \cdot A}} \]

   if 2.0000000000000001e130 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\sqrt{V \cdot \frac{A}{\ell}} \cdot \frac{c0}{V}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ t_1 := \sqrt{V\_m \cdot \frac{A\_m}{l\_m}} \cdot \frac{c0}{V\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+130}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{l\_m \cdot V\_m} \cdot A\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m))))
        (t_1 (* (sqrt (* V_m (/ A_m l_m))) (/ c0 V_m))))
   (if (<= t_0 2e-157)
     t_1
     (if (<= t_0 2e+130) (* c0 (sqrt (* (/ 1.0 (* l_m V_m)) A_m))) t_1))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double t_1 = sqrt((V_m * (A_m / l_m))) * (c0 / V_m);
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = t_1;
	} else if (t_0 <= 2e+130) {
		tmp = c0 * sqrt(((1.0 / (l_m * V_m)) * A_m));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    t_1 = sqrt((v_m * (a_m / l_m))) * (c0 / v_m)
    if (t_0 <= 2d-157) then
        tmp = t_1
    else if (t_0 <= 2d+130) then
        tmp = c0 * sqrt(((1.0d0 / (l_m * v_m)) * a_m))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double t_1 = Math.sqrt((V_m * (A_m / l_m))) * (c0 / V_m);
	double tmp;
	if (t_0 <= 2e-157) {
		tmp = t_1;
	} else if (t_0 <= 2e+130) {
		tmp = c0 * Math.sqrt(((1.0 / (l_m * V_m)) * A_m));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	t_1 = math.sqrt((V_m * (A_m / l_m))) * (c0 / V_m)
	tmp = 0
	if t_0 <= 2e-157:
		tmp = t_1
	elif t_0 <= 2e+130:
		tmp = c0 * math.sqrt(((1.0 / (l_m * V_m)) * A_m))
	else:
		tmp = t_1
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	t_1 = Float64(sqrt(Float64(V_m * Float64(A_m / l_m))) * Float64(c0 / V_m))
	tmp = 0.0
	if (t_0 <= 2e-157)
		tmp = t_1;
	elseif (t_0 <= 2e+130)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / Float64(l_m * V_m)) * A_m)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	t_1 = sqrt((V_m * (A_m / l_m))) * (c0 / V_m);
	tmp = 0.0;
	if (t_0 <= 2e-157)
		tmp = t_1;
	elseif (t_0 <= 2e+130)
		tmp = c0 * sqrt(((1.0 / (l_m * V_m)) * A_m));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(V$95$m * N[(A$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(c0 / V$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-157], t$95$1, If[LessEqual[t$95$0, 2e+130], N[(c0 * N[Sqrt[N[(N[(1.0 / N[(l$95$m * V$95$m), $MachinePrecision]), $MachinePrecision] * A$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.99999999999999989e-157 or 2.0000000000000001e130 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   4. Taylor expanded in V around 0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A \cdot V}{\ell}}}{V}} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\sqrt{V \cdot \frac{A}{\ell}} \cdot \frac{c0}{V}} \]

   if 1.99999999999999989e-157 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 2.0000000000000001e130

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. mult-flip N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell}} \cdot A} \]

      7. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

      8. lower-*.f64 72.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot V}} \cdot A} \]

   3. Applied rewrites 72.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot V} \cdot A}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} \mathbf{if}\;A\_m \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A\_m}{V\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A\_m}{l\_m}}{V\_m}}\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (if (<= A_m 1.1e-88)
   (* c0 (sqrt (/ (/ A_m V_m) l_m)))
   (* c0 (sqrt (/ (/ A_m l_m) V_m)))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double tmp;
	if (A_m <= 1.1e-88) {
		tmp = c0 * sqrt(((A_m / V_m) / l_m));
	} else {
		tmp = c0 * sqrt(((A_m / l_m) / V_m));
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: tmp
    if (a_m <= 1.1d-88) then
        tmp = c0 * sqrt(((a_m / v_m) / l_m))
    else
        tmp = c0 * sqrt(((a_m / l_m) / v_m))
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double tmp;
	if (A_m <= 1.1e-88) {
		tmp = c0 * Math.sqrt(((A_m / V_m) / l_m));
	} else {
		tmp = c0 * Math.sqrt(((A_m / l_m) / V_m));
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	tmp = 0
	if A_m <= 1.1e-88:
		tmp = c0 * math.sqrt(((A_m / V_m) / l_m))
	else:
		tmp = c0 * math.sqrt(((A_m / l_m) / V_m))
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	tmp = 0.0
	if (A_m <= 1.1e-88)
		tmp = Float64(c0 * sqrt(Float64(Float64(A_m / V_m) / l_m)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A_m / l_m) / V_m)));
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	tmp = 0.0;
	if (A_m <= 1.1e-88)
		tmp = c0 * sqrt(((A_m / V_m) / l_m));
	else
		tmp = c0 * sqrt(((A_m / l_m) / V_m));
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := If[LessEqual[A$95$m, 1.1e-88], N[(c0 * N[Sqrt[N[(N[(A$95$m / V$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A$95$m / l$95$m), $MachinePrecision] / V$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 1.10000000000000002e-88

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 1.10000000000000002e-88 < A

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      4. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

      5. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

      6. lower-/.f64 73.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

   3. Applied rewrites 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A\_m}{V\_m \cdot l\_m}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A\_m}{V\_m}}{l\_m}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;c0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ A_m (* V_m l_m))))
        (t_1 (* c0 (sqrt (/ (/ A_m V_m) l_m)))))
   (if (<= t_0 5e-10) t_1 (if (<= t_0 5e+146) (* c0 t_0) t_1))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = sqrt((A_m / (V_m * l_m)));
	double t_1 = c0 * sqrt(((A_m / V_m) / l_m));
	double tmp;
	if (t_0 <= 5e-10) {
		tmp = t_1;
	} else if (t_0 <= 5e+146) {
		tmp = c0 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a_m / (v_m * l_m)))
    t_1 = c0 * sqrt(((a_m / v_m) / l_m))
    if (t_0 <= 5d-10) then
        tmp = t_1
    else if (t_0 <= 5d+146) then
        tmp = c0 * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	double t_0 = Math.sqrt((A_m / (V_m * l_m)));
	double t_1 = c0 * Math.sqrt(((A_m / V_m) / l_m));
	double tmp;
	if (t_0 <= 5e-10) {
		tmp = t_1;
	} else if (t_0 <= 5e+146) {
		tmp = c0 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	t_0 = math.sqrt((A_m / (V_m * l_m)))
	t_1 = c0 * math.sqrt(((A_m / V_m) / l_m))
	tmp = 0
	if t_0 <= 5e-10:
		tmp = t_1
	elif t_0 <= 5e+146:
		tmp = c0 * t_0
	else:
		tmp = t_1
	return tmp

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	t_0 = sqrt(Float64(A_m / Float64(V_m * l_m)))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A_m / V_m) / l_m)))
	tmp = 0.0
	if (t_0 <= 5e-10)
		tmp = t_1;
	elseif (t_0 <= 5e+146)
		tmp = Float64(c0 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp_2 = code(c0, A_m, V_m, l_m)
	t_0 = sqrt((A_m / (V_m * l_m)));
	t_1 = c0 * sqrt(((A_m / V_m) / l_m));
	tmp = 0.0;
	if (t_0 <= 5e-10)
		tmp = t_1;
	elseif (t_0 <= 5e+146)
		tmp = c0 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := Block[{t$95$0 = N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A$95$m / V$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-10], t$95$1, If[LessEqual[t$95$0, 5e+146], N[(c0 * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.00000000000000031e-10 or 4.9999999999999999e146 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   3. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 5.00000000000000031e-10 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 4.9999999999999999e146

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} A_m = \left|A\right| \\ V_m = \left|V\right| \\ l_m = \left|\ell\right| \\ [c0, A_m, V_m, l_m] = \mathsf{sort}([c0, A_m, V_m, l_m])\\ \\ c0 \cdot \sqrt{\frac{A\_m}{V\_m \cdot l\_m}} \end{array} \]

FPCore

A_m = (fabs.f64 A)
V_m = (fabs.f64 V)
l_m = (fabs.f64 l)
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
(FPCore (c0 A_m V_m l_m) :precision binary64 (* c0 (sqrt (/ A_m (* V_m l_m)))))

C

A_m = fabs(A);
V_m = fabs(V);
l_m = fabs(l);
assert(c0 < A_m && A_m < V_m && V_m < l_m);
double code(double c0, double A_m, double V_m, double l_m) {
	return c0 * sqrt((A_m / (V_m * l_m)));
}

Fortran

A_m =     private
V_m =     private
l_m =     private
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
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(c0, a_m, v_m, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a_m
    real(8), intent (in) :: v_m
    real(8), intent (in) :: l_m
    code = c0 * sqrt((a_m / (v_m * l_m)))
end function

Java

A_m = Math.abs(A);
V_m = Math.abs(V);
l_m = Math.abs(l);
assert c0 < A_m && A_m < V_m && V_m < l_m;
public static double code(double c0, double A_m, double V_m, double l_m) {
	return c0 * Math.sqrt((A_m / (V_m * l_m)));
}

Python

A_m = math.fabs(A)
V_m = math.fabs(V)
l_m = math.fabs(l)
[c0, A_m, V_m, l_m] = sort([c0, A_m, V_m, l_m])
def code(c0, A_m, V_m, l_m):
	return c0 * math.sqrt((A_m / (V_m * l_m)))

Julia

A_m = abs(A)
V_m = abs(V)
l_m = abs(l)
c0, A_m, V_m, l_m = sort([c0, A_m, V_m, l_m])
function code(c0, A_m, V_m, l_m)
	return Float64(c0 * sqrt(Float64(A_m / Float64(V_m * l_m))))
end

MATLAB

A_m = abs(A);
V_m = abs(V);
l_m = abs(l);
c0, A_m, V_m, l_m = num2cell(sort([c0, A_m, V_m, l_m])){:}
function tmp = code(c0, A_m, V_m, l_m)
	tmp = c0 * sqrt((A_m / (V_m * l_m)));
end

Wolfram

A_m = N[Abs[A], $MachinePrecision]
V_m = N[Abs[V], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: c0, A_m, V_m, and l_m should be sorted in increasing order before calling this function.
code[c0_, A$95$m_, V$95$m_, l$95$m_] := N[(c0 * N[Sqrt[N[(A$95$m / N[(V$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))