Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.2% → 91.4%

Time: 4.5s

Alternatives: 15

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: 74.2% 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: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\left(-\frac{c0}{V}\right) \cdot \sqrt{\frac{-A}{\frac{-\ell}{V}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e-302)
   (* c0 (/ (sqrt (- A)) (* (sqrt (- V)) (sqrt l))))
   (if (<= (* V l) 0.0)
     (* (- (/ c0 V)) (sqrt (/ (- A) (/ (- l) V))))
     (/ c0 (/ (sqrt (* V l)) (sqrt A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-302) {
		tmp = c0 * (sqrt(-A) / (sqrt(-V) * sqrt(l)));
	} else if ((V * l) <= 0.0) {
		tmp = -(c0 / V) * sqrt((-A / (-l / V)));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if ((v * l) <= (-5d-302)) then
        tmp = c0 * (sqrt(-a) / (sqrt(-v) * sqrt(l)))
    else if ((v * l) <= 0.0d0) then
        tmp = -(c0 / v) * sqrt((-a / (-l / v)))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-302) {
		tmp = c0 * (Math.sqrt(-A) / (Math.sqrt(-V) * Math.sqrt(l)));
	} else if ((V * l) <= 0.0) {
		tmp = -(c0 / V) * Math.sqrt((-A / (-l / V)));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e-302:
		tmp = c0 * (math.sqrt(-A) / (math.sqrt(-V) * math.sqrt(l)))
	elif (V * l) <= 0.0:
		tmp = -(c0 / V) * math.sqrt((-A / (-l / V)))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e-302)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / Float64(sqrt(Float64(-V)) * sqrt(l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(Float64(-Float64(c0 / V)) * sqrt(Float64(Float64(-A) / Float64(Float64(-l) / V))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e-302)
		tmp = c0 * (sqrt(-A) / (sqrt(-V) * sqrt(l)));
	elseif ((V * l) <= 0.0)
		tmp = -(c0 / V) * sqrt((-A / (-l / V)));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e-302], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[((-N[(c0 / V), $MachinePrecision]) * N[Sqrt[N[((-A) / N[((-l) / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-sqrt.f64 49.2

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

   5. Applied rewrites 49.2%

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

   if -5.00000000000000033e-302 < (*.f64 V l) < -0.0

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in V around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left(A\right)}}{V \cdot \sqrt{-1 \cdot \frac{\ell}{V}}}} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-undiv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

      9. associate-/l/ N/A

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

      10. sqrt-div N/A

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

      11. mul-1-neg N/A

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

      12. times-frac N/A

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

      13. distribute-lft-neg-in N/A

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

   6. Applied rewrites 49.4%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 2: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+263}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}} \cdot c0}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{\ell}{V} \cdot A} \cdot \frac{c0}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+263)
   (/ (* (sqrt (/ (- A) l)) c0) (sqrt (- V)))
   (if (<= (* V l) -5e-302)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- l) V))))
     (if (<= (* V l) 0.0)
       (* (sqrt (* (/ l V) A)) (/ c0 l))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+263) {
		tmp = (sqrt((-A / l)) * c0) / sqrt(-V);
	} else if ((V * l) <= -5e-302) {
		tmp = c0 * (sqrt(-A) / sqrt((-l * V)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt(((l / V) * A)) * (c0 / l);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if ((v * l) <= (-1d+263)) then
        tmp = (sqrt((-a / l)) * c0) / sqrt(-v)
    else if ((v * l) <= (-5d-302)) then
        tmp = c0 * (sqrt(-a) / sqrt((-l * v)))
    else if ((v * l) <= 0.0d0) then
        tmp = sqrt(((l / v) * a)) * (c0 / l)
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+263) {
		tmp = (Math.sqrt((-A / l)) * c0) / Math.sqrt(-V);
	} else if ((V * l) <= -5e-302) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-l * V)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt(((l / V) * A)) * (c0 / l);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+263:
		tmp = (math.sqrt((-A / l)) * c0) / math.sqrt(-V)
	elif (V * l) <= -5e-302:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-l * V)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt(((l / V) * A)) * (c0 / l)
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+263)
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) * c0) / sqrt(Float64(-V)));
	elseif (Float64(V * l) <= -5e-302)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(l / V) * A)) * Float64(c0 / l));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+263)
		tmp = (sqrt((-A / l)) * c0) / sqrt(-V);
	elseif ((V * l) <= -5e-302)
		tmp = c0 * (sqrt(-A) / sqrt((-l * V)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt(((l / V) * A)) * (c0 / l);
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+263], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-302], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[Sqrt[N[(N[(l / V), $MachinePrecision] * A), $MachinePrecision]], $MachinePrecision] * N[(c0 / l), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.00000000000000002e263

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{2}}} \cdot c0 \]

      7. sqr-pow N/A

         \[\leadsto \color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot c0 \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      11. lift-/.f64 N/A

          \[\leadsto {\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      12. *-commutative N/A

          \[\leadsto {\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto {\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\color{blue}{\frac{1}{4}}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      15. lower-*.f64 N/A

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot \color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right)} \]

      16. lower-pow.f64 N/A

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot c0\right) \]

      17. lift-/.f64 N/A

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot \left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      18. *-commutative N/A

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      19. lower-*.f64 N/A

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot c0\right) \]

      20. metadata-eval 74.0

          \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{0.25} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\color{blue}{0.25}} \cdot c0\right) \]

   3. Applied rewrites 74.0%

      \[\leadsto \color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{0.25} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{0.25} \cdot c0\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right) \]

      3. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{A}{\ell \cdot V}\right)}}^{\frac{1}{4}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right) \]

      4. *-commutative N/A

         \[\leadsto {\left(\frac{A}{\color{blue}{V \cdot \ell}}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right) \]

      6. sqr-pow N/A

         \[\leadsto \color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \color{blue}{\left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left(\color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}}} \cdot c0\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\frac{1}{4}} \cdot c0\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({\color{blue}{\left(\frac{A}{\ell \cdot V}\right)}}^{\frac{1}{4}} \cdot c0\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)\right)} \]

      13. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto {\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto {\left(\frac{A}{\color{blue}{V \cdot \ell}}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{\frac{1}{8}}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

   5. Applied rewrites 73.9%

      \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.125} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{0.125} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25} \cdot c0\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{\color{blue}{V \cdot \ell}}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left(\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{\color{blue}{V \cdot \ell}}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left(\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}}} \cdot c0\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{\color{blue}{V \cdot \ell}}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\frac{1}{4}} \cdot c0\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}} \cdot c0\right)\right) \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}}} \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left({\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{4}}\right) \cdot c0\right)} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{\frac{1}{8}} \]

   7. Applied rewrites 74.1%

      \[\leadsto \color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{0.375} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{0.125} \cdot c0\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot c0\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot c0\right) \]

      3. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\frac{3}{8}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot c0\right) \]

      4. lift-/.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{A}{\ell \cdot V}\right)}}^{\frac{3}{8}} \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot c0\right) \]

      5. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}} \cdot \color{blue}{\left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot c0\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}} \cdot \left(\color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}}} \cdot c0\right) \]

      7. lift-*.f64 N/A

         \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}} \cdot \left({\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\frac{1}{8}} \cdot c0\right) \]

      8. lift-/.f64 N/A

         \[\leadsto {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}} \cdot \left({\color{blue}{\left(\frac{A}{\ell \cdot V}\right)}}^{\frac{1}{8}} \cdot c0\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot c0\right) \cdot {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(c0 \cdot {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}}\right)} \cdot {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}} \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{c0 \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}}\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto c0 \cdot \left({\color{blue}{\left(\frac{A}{\ell \cdot V}\right)}}^{\frac{1}{8}} \cdot {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto c0 \cdot \left({\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\frac{1}{8}} \cdot {\left(\frac{A}{\ell \cdot V}\right)}^{\frac{3}{8}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto c0 \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot {\color{blue}{\left(\frac{A}{\ell \cdot V}\right)}}^{\frac{3}{8}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto c0 \cdot \left({\left(\frac{A}{\ell \cdot V}\right)}^{\frac{1}{8}} \cdot {\left(\frac{A}{\color{blue}{\ell \cdot V}}\right)}^{\frac{3}{8}}\right) \]

      16. pow-add N/A

          \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{\ell \cdot V}\right)}^{\left(\frac{1}{8} + \frac{3}{8}\right)}} \]

   9. Applied rewrites 61.3%

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

   if -1.00000000000000002e263 < (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 42.3

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

   3. Applied rewrites 42.3%

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

   if -5.00000000000000033e-302 < (*.f64 V l) < -0.0

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.9

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

   3. Applied rewrites 73.9%

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

   4. Taylor expanded in l around 0

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

      1. sqrt-div N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 48.7

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

   6. Applied rewrites 48.7%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 3: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{\ell}{V} \cdot A} \cdot \frac{c0}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -5e-302)
     (* c0 (/ (sqrt (- A)) (sqrt (* (- l) V))))
     (if (<= (* V l) 0.0)
       (* (sqrt (* (/ l V) A)) (/ c0 l))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -5e-302) {
		tmp = c0 * (sqrt(-A) / sqrt((-l * V)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt(((l / V) * A)) * (c0 / l);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -5e-302) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-l * V)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt(((l / V) * A)) * (c0 / l);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -5e-302:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-l * V)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt(((l / V) * A)) * (c0 / l)
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -5e-302)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(l / V) * A)) * Float64(c0 / l));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -5e-302)
		tmp = c0 * (sqrt(-A) / sqrt((-l * V)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt(((l / V) * A)) * (c0 / l);
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-302], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[Sqrt[N[(N[(l / V), $MachinePrecision] * A), $MachinePrecision]], $MachinePrecision] * N[(c0 / l), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. mul-1-neg N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 42.3

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

   3. Applied rewrites 42.3%

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

   if -5.00000000000000033e-302 < (*.f64 V l) < -0.0

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.9

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

   3. Applied rewrites 73.9%

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

   4. Taylor expanded in l around 0

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

      1. sqrt-div N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 48.7

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

   6. Applied rewrites 48.7%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 4: 90.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (* c0 (/ (sqrt (- A)) (* (sqrt (- V)) (sqrt l))))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * (sqrt(-A) / (sqrt(-V) * sqrt(l)));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = c0 * (sqrt(-a) / (sqrt(-v) * sqrt(l)))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * (Math.sqrt(-A) / (Math.sqrt(-V) * Math.sqrt(l)));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = c0 * (math.sqrt(-A) / (math.sqrt(-V) * math.sqrt(l)))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / Float64(sqrt(Float64(-V)) * sqrt(l))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = c0 * (sqrt(-A) / (sqrt(-V) * sqrt(l)));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-sqrt.f64 49.2

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

   5. Applied rewrites 49.2%

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

   if -1.999999999999994e-310 < A

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 5: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\ell}{V} \cdot A} \cdot \frac{c0}{\ell}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (* (/ l V) A)) (/ c0 l))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-302)
       (* c0 (/ (sqrt (- A)) (sqrt (* (- l) V))))
       (if (<= (* V l) 0.0) t_0 (/ c0 (/ (sqrt (* V l)) (sqrt A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(((l / V) * A)) * (c0 / l);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-302) {
		tmp = c0 * (sqrt(-A) / sqrt((-l * V)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(((l / V) * A)) * (c0 / l);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-302) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-l * V)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(((l / V) * A)) * (c0 / l)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-302:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-l * V)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(Float64(l / V) * A)) * Float64(c0 / l))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-302)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt(((l / V) * A)) * (c0 / l);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-302)
		tmp = c0 * (sqrt(-A) / sqrt((-l * V)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(l / V), $MachinePrecision] * A), $MachinePrecision]], $MachinePrecision] * N[(c0 / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-302], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -5.00000000000000033e-302 < (*.f64 V l) < -0.0

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.9

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

   3. Applied rewrites 73.9%

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

   4. Taylor expanded in l around 0

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

      1. sqrt-div N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 48.7

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

   6. Applied rewrites 48.7%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 42.3

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

   3. Applied rewrites 42.3%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 6: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 2.9 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A 2.9e-290)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-290) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if (a <= 2.9d-290) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-290) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= 2.9e-290:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= 2.9e-290)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= 2.9e-290)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, 2.9e-290], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 2.89999999999999994e-290

   1. Initial program 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 63.3

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

   3. Applied rewrites 63.3%

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

   if 2.89999999999999994e-290 < A

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 2.9 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A 2.9e-290)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-290) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if (a <= 2.9d-290) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= 2.9e-290) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= 2.9e-290:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= 2.9e-290)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= 2.9e-290)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, 2.9e-290], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 2.89999999999999994e-290

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if 2.89999999999999994e-290 < A

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 42.2

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

   9. Applied rewrites 42.2%

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

Alternative 8: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-310)
   (/ (* (sqrt A) c0) (sqrt (* l V)))
   (/ c0 (* (sqrt l) (sqrt (/ V A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	} else {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	} else {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	else:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	else
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	else
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{{A}^{\frac{1}{2}}}}{\sqrt{V \cdot \ell}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot {A}^{\frac{1}{2}}}{\sqrt{V \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot {A}^{\frac{1}{2}}}{\sqrt{V \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{A}^{\frac{1}{2}} \cdot c0}}{\sqrt{V \cdot \ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{A}^{\frac{1}{2}} \cdot c0}}{\sqrt{V \cdot \ell}} \]

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.8

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

   3. Applied rewrites 40.8%

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

   if -4.999999999999985e-310 < l

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 63.6

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

   9. Applied rewrites 63.6%

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

Alternative 9: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 2 \cdot 10^{-297}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 2e-297)
   (/ c0 (sqrt (* (/ l A) V)))
   (/ (* (sqrt A) c0) (sqrt (* l V)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 2e-297) {
		tmp = c0 / sqrt(((l / A) * V));
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if ((v * l) <= 2d-297) then
        tmp = c0 / sqrt(((l / a) * v))
    else
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 2e-297) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 2e-297:
		tmp = c0 / math.sqrt(((l / A) * V))
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 2e-297)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 2e-297)
		tmp = c0 / sqrt(((l / A) * V));
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 2e-297], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 2.00000000000000008e-297

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.5

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

   9. Applied rewrites 74.5%

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

   if 2.00000000000000008e-297 < (*.f64 V l)

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{{A}^{\frac{1}{2}}}}{\sqrt{V \cdot \ell}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot {A}^{\frac{1}{2}}}{\sqrt{V \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot {A}^{\frac{1}{2}}}{\sqrt{V \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{A}^{\frac{1}{2}} \cdot c0}}{\sqrt{V \cdot \ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{A}^{\frac{1}{2}} \cdot c0}}{\sqrt{V \cdot \ell}} \]

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.8

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

   3. Applied rewrites 40.8%

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

Alternative 10: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 2e-173)
   (/ c0 (sqrt (* (/ l A) V)))
   (* (sqrt A) (/ c0 (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 2e-173) {
		tmp = c0 / sqrt(((l / A) * V));
	} else {
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if ((v * l) <= 2d-173) then
        tmp = c0 / sqrt(((l / a) * v))
    else
        tmp = sqrt(a) * (c0 / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 2e-173) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 2e-173:
		tmp = c0 / math.sqrt(((l / A) * V))
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 2e-173)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 2e-173)
		tmp = c0 / sqrt(((l / A) * V));
	else
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 2e-173], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 2.0000000000000001e-173

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.5

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

   9. Applied rewrites 74.5%

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

   if 2.0000000000000001e-173 < (*.f64 V l)

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{A}^{\frac{1}{2}}}}{\sqrt{V \cdot \ell}} \cdot c0 \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{A}^{\frac{1}{2}} \cdot c0}{\sqrt{V \cdot \ell}}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{{A}^{\frac{1}{2}} \cdot \frac{c0}{\sqrt{V \cdot \ell}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{{A}^{\frac{1}{2}} \cdot \frac{c0}{\sqrt{V \cdot \ell}}} \]

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 41.0

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

   3. Applied rewrites 41.0%

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

Alternative 11: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 1e+124) t_0 (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 1d+124) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 1e+124:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 1e+124)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 1e+124)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+124], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 74.4

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

   3. Applied rewrites 74.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999948e123

   1. Initial program 74.2%

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

   if 9.99999999999999948e123 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.9

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

   3. Applied rewrites 73.9%

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

Alternative 12: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 1e+124) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+124) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+124:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+124)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+124)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+124], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 9.99999999999999948e123 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.9

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

   3. Applied rewrites 73.9%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999948e123

   1. Initial program 74.2%

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

Alternative 13: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))) (t_1 (/ c0 (sqrt (* (/ V A) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2e+297) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double t_1 = c0 / sqrt(((V / A) * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+297) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    t_1 = c0 / sqrt(((v / a) * l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+297) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double t_1 = c0 / Math.sqrt(((V / A) * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+297) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	t_1 = c0 / math.sqrt(((V / A) * l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+297:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	t_1 = Float64(c0 / sqrt(Float64(Float64(V / A) * l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+297)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	t_1 = c0 / sqrt(((V / A) * l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+297)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+297], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 2e297 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.0

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

   9. Applied rewrites 74.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e297

   1. Initial program 74.2%

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

Alternative 14: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-232)
   (/ c0 (sqrt (* (/ l A) V)))
   (* c0 (sqrt (/ A (* V l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-232) {
		tmp = c0 / sqrt(((l / A) * V));
	} else {
		tmp = c0 * sqrt((A / (V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l 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, 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
    real(8) :: tmp
    if ((v * l) <= 5d-232) then
        tmp = c0 / sqrt(((l / a) * v))
    else
        tmp = c0 * sqrt((a / (v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-232) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else {
		tmp = c0 * Math.sqrt((A / (V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 5e-232:
		tmp = c0 / math.sqrt(((l / A) * V))
	else:
		tmp = c0 * math.sqrt((A / (V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-232)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	else
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 5e-232)
		tmp = c0 / sqrt(((l / A) * V));
	else
		tmp = c0 * sqrt((A / (V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-232], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 4.9999999999999999e-232

   1. Initial program 74.2%

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

   2. Taylor expanded in A around inf

      \[\leadsto \color{blue}{A \cdot \left(c0 \cdot \sqrt{\frac{1}{A \cdot \left(V \cdot \ell\right)}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.6

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in A around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.5

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

   9. Applied rewrites 74.5%

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

   if 4.9999999999999999e-232 < (*.f64 V l)

   1. Initial program 74.2%

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

Alternative 15: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

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

C

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

Fortran

NOTE: c0, A, V, and l 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, 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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

Reproduce

herbie shell --seed 2025138 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))