Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 89.5%

Time: 6.7s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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: 89.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := {\left(-A\right)}^{0.25}\\ \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\frac{t\_0}{\sqrt{-V}} \cdot \frac{t\_0}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (pow (- A) 0.25)))
   (if (<= A -5e-310)
     (* c0 (* (/ t_0 (sqrt (- V))) (/ t_0 (sqrt l))))
     (/ (* (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 t_0 = pow(-A, 0.25);
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((t_0 / sqrt(-V)) * (t_0 / sqrt(l)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -a ** 0.25d0
    if (a <= (-5d-310)) then
        tmp = c0 * ((t_0 / sqrt(-v)) * (t_0 / sqrt(l)))
    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 t_0 = Math.pow(-A, 0.25);
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((t_0 / Math.sqrt(-V)) * (t_0 / Math.sqrt(l)));
	} 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):
	t_0 = math.pow(-A, 0.25)
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * ((t_0 / math.sqrt(-V)) * (t_0 / math.sqrt(l)))
	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)
	t_0 = Float64(-A) ^ 0.25
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(Float64(t_0 / sqrt(Float64(-V))) * Float64(t_0 / sqrt(l))));
	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)
	t_0 = -A ^ 0.25;
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * ((t_0 / sqrt(-V)) * (t_0 / sqrt(l)));
	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_] := Block[{t$95$0 = N[Power[(-A), 0.25], $MachinePrecision]}, If[LessEqual[A, -5e-310], N[(c0 * N[(N[(t$95$0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $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 A < -4.999999999999985e-310

   1. Initial program 64.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. sqr-pow N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. metadata-eval N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 85.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

Alternative 2: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-212} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (or (<= t_0 5e-212) (not (<= t_0 5e+304)))
     (/ c0 (sqrt (* (/ V A) l)))
     t_0)))

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 <= 5e-212) || !(t_0 <= 5e+304)) {
		tmp = c0 / sqrt(((V / A) * l));
	} else {
		tmp = t_0;
	}
	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 <= 5d-212) .or. (.not. (t_0 <= 5d+304))) then
        tmp = c0 / sqrt(((v / a) * l))
    else
        tmp = t_0
    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 <= 5e-212) || !(t_0 <= 5e+304)) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else {
		tmp = t_0;
	}
	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 <= 5e-212) or not (t_0 <= 5e+304):
		tmp = c0 / math.sqrt(((V / A) * l))
	else:
		tmp = t_0
	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 <= 5e-212) || !(t_0 <= 5e+304))
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	else
		tmp = t_0;
	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 <= 5e-212) || ~((t_0 <= 5e+304)))
		tmp = c0 / sqrt(((V / A) * l));
	else
		tmp = t_0;
	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[Or[LessEqual[t$95$0, 5e-212], N[Not[LessEqual[t$95$0, 5e+304]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000043e-212 or 4.9999999999999997e304 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 57.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 58.0

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

   7. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 58.0

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

   9. Applied rewrites 58.0%

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

   11. Applied rewrites 67.7%

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

   if 5.00000000000000043e-212 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999997e304

   1. Initial program 98.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-212} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+295}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (or (<= t_0 0.0) (not (<= t_0 1e+295)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

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) || !(t_0 <= 1e+295)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	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) .or. (.not. (t_0 <= 1d+295))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    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) || !(t_0 <= 1e+295)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	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) or not (t_0 <= 1e+295):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	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) || !(t_0 <= 1e+295))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	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) || ~((t_0 <= 1e+295)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+295]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 9.9999999999999998e294 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 67.1

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

   4. Applied rewrites 67.1%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999998e294

   1. Initial program 98.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+295}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-159}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+136}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A (* V l)))))
   (if (<= t_0 5e-159)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+136)
       (/ c0 (sqrt (/ (* l V) A)))
       (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-159) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+136) {
		tmp = c0 / sqrt(((l * V) / A));
	} else {
		tmp = c0 / sqrt(((V / A) * 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 = sqrt((a / (v * l)))
    if (t_0 <= 5d-159) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+136) then
        tmp = c0 / sqrt(((l * v) / a))
    else
        tmp = c0 / sqrt(((v / a) * 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 = Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-159) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+136) {
		tmp = c0 / Math.sqrt(((l * V) / A));
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-159:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+136:
		tmp = c0 / math.sqrt(((l * V) / A))
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / Float64(V * l)))
	tmp = 0.0
	if (t_0 <= 5e-159)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+136)
		tmp = Float64(c0 / sqrt(Float64(Float64(l * V) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * 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 = sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-159)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+136)
		tmp = c0 / sqrt(((l * V) / A));
	else
		tmp = c0 / sqrt(((V / A) * 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[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-159], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+136], N[(c0 / N[Sqrt[N[(N[(l * V), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.00000000000000032e-159

   1. Initial program 29.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 48.7

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

   4. Applied rewrites 48.7%

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

   if 5.00000000000000032e-159 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 1.00000000000000006e136

   1. Initial program 99.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.00000000000000006e136 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 34.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 35.0

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

   7. Applied rewrites 35.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 35.0

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

   9. Applied rewrites 35.0%

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

   11. Applied rewrites 54.1%

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

Alternative 5: 89.5% accurate, 0.5× speedup

Math

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

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 (<= A -5e-310)
   (/ c0 (* (sqrt l) (/ (sqrt (- V)) (sqrt (- A)))))
   (/ (* (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 (A <= -5e-310) {
		tmp = c0 / (sqrt(l) * (sqrt(-V) / sqrt(-A)));
	} 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 (a <= (-5d-310)) then
        tmp = c0 / (sqrt(l) * (sqrt(-v) / sqrt(-a)))
    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 (A <= -5e-310) {
		tmp = c0 / (Math.sqrt(l) * (Math.sqrt(-V) / Math.sqrt(-A)));
	} 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 A <= -5e-310:
		tmp = c0 / (math.sqrt(l) * (math.sqrt(-V) / math.sqrt(-A)))
	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 (A <= -5e-310)
		tmp = Float64(c0 / Float64(sqrt(l) * Float64(sqrt(Float64(-V)) / sqrt(Float64(-A)))));
	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 (A <= -5e-310)
		tmp = c0 / (sqrt(l) * (sqrt(-V) / sqrt(-A)));
	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[A, -5e-310], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $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 A < -4.999999999999985e-310

   1. Initial program 64.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.5

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

   7. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 64.5

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

   9. Applied rewrites 64.5%

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

   11. Applied rewrites 52.1%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 85.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \frac{\sqrt{-V}}{\sqrt{-A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.5× speedup

Math

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

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 (<= A -5e-310)
   (/ (* (sqrt (- A)) c0) (* (sqrt (- V)) (sqrt l)))
   (/ (* (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 (A <= -5e-310) {
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	} 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 (a <= (-5d-310)) then
        tmp = (sqrt(-a) * c0) / (sqrt(-v) * sqrt(l))
    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 (A <= -5e-310) {
		tmp = (Math.sqrt(-A) * c0) / (Math.sqrt(-V) * Math.sqrt(l));
	} 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 A <= -5e-310:
		tmp = (math.sqrt(-A) * c0) / (math.sqrt(-V) * math.sqrt(l))
	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 (A <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(sqrt(Float64(-V)) * sqrt(l)));
	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 (A <= -5e-310)
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	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[A, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $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 A < -4.999999999999985e-310

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. associate-*r/ N/A

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

      19. lift-/.f64 N/A

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

      20. sqrt-div N/A

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

      21. lift-sqrt.f64 N/A

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

      22. associate-*r/ N/A

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

      23. associate-/r* N/A

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

      24. frac-times N/A

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

   6. Applied rewrites 51.7%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 85.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{-\ell} \cdot \sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= l -4e-310)
   (/ (* (sqrt A) c0) (* (sqrt (- l)) (sqrt (- V))))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (sqrt(A) * c0) / (sqrt(-l) * sqrt(-V));
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(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 (l <= (-4d-310)) then
        tmp = (sqrt(a) * c0) / (sqrt(-l) * sqrt(-v))
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(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 (l <= -4e-310) {
		tmp = (Math.sqrt(A) * c0) / (Math.sqrt(-l) * Math.sqrt(-V));
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -4e-310:
		tmp = (math.sqrt(A) * c0) / (math.sqrt(-l) * math.sqrt(-V))
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(sqrt(A) * c0) / Float64(sqrt(Float64(-l)) * sqrt(Float64(-V))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(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 (l <= -4e-310)
		tmp = (sqrt(A) * c0) / (sqrt(-l) * sqrt(-V));
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(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[l, -4e-310], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-l)], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 38.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 38.3

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

   4. Applied rewrites 38.3%

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

      1. lift-sqrt.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 46.7

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

   6. Applied rewrites 46.7%

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

   if -3.999999999999988e-310 < l

   1. Initial program 69.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 70.4

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 79.5%

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

Alternative 8: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{-\ell} \cdot \sqrt{-V}} \cdot \sqrt{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= l -4e-310)
   (* (/ c0 (* (sqrt (- l)) (sqrt (- V)))) (sqrt A))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -4e-310) {
		tmp = (c0 / (sqrt(-l) * sqrt(-V))) * sqrt(A);
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(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 (l <= (-4d-310)) then
        tmp = (c0 / (sqrt(-l) * sqrt(-v))) * sqrt(a)
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(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 (l <= -4e-310) {
		tmp = (c0 / (Math.sqrt(-l) * Math.sqrt(-V))) * Math.sqrt(A);
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -4e-310:
		tmp = (c0 / (math.sqrt(-l) * math.sqrt(-V))) * math.sqrt(A)
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(c0 / Float64(sqrt(Float64(-l)) * sqrt(Float64(-V)))) * sqrt(A));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(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 (l <= -4e-310)
		tmp = (c0 / (sqrt(-l) * sqrt(-V))) * sqrt(A);
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(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[l, -4e-310], N[(N[(c0 / N[(N[Sqrt[(-l)], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 67.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 39.1

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

   4. Applied rewrites 39.1%

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

      1. lift-sqrt.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 46.6

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

   6. Applied rewrites 46.6%

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

   if -3.999999999999988e-310 < l

   1. Initial program 69.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 70.4

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 79.5%

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

Alternative 9: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

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) 0.0)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (/ (* (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) <= 0.0) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} 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) <= 0.0d0) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    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) <= 0.0) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} 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) <= 0.0:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	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) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	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) <= 0.0)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	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], 0.0], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $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) < -0.0

   1. Initial program 60.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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. pow1/2 N/A

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

      4. pow-to-exp N/A

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

      5. sinh-+-cosh-rev N/A

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

      6. flip-+ N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. exp-neg N/A

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

      13. pow-to-exp N/A

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

      14. pow1/2 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in A around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 60.7

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

   7. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 60.7

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

   9. Applied rewrites 60.7%

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

   11. Applied rewrites 41.5%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   4. Applied rewrites 91.9%

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

Alternative 10: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

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) 0.0)
   (* (/ c0 (sqrt l)) (sqrt (/ 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) <= 0.0) {
		tmp = (c0 / sqrt(l)) * sqrt((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) <= 0.0d0) then
        tmp = (c0 / sqrt(l)) * sqrt((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) <= 0.0) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((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) <= 0.0:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((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) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(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) <= 0.0)
		tmp = (c0 / sqrt(l)) * sqrt((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], 0.0], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / 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) < -0.0

   1. Initial program 60.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 41.0

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

   4. Applied rewrites 41.0%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   4. Applied rewrites 91.9%

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

Alternative 11: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

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) 0.0)
   (* c0 (sqrt (/ (/ A l) 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) <= 0.0) {
		tmp = c0 * sqrt(((A / l) / 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) <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / 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) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / 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) <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / 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) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / 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) <= 0.0)
		tmp = c0 * sqrt(((A / l) / 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], 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $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) < -0.0

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 71.9

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

   4. Applied rewrites 71.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. frac-2neg N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 91.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.9

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

   4. Applied rewrites 91.9%

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

Alternative 12: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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) 0.0)
   (* c0 (sqrt (/ (/ A l) V)))
   (* (/ c0 (sqrt (* l V))) (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) <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = (c0 / sqrt((l * V))) * 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) <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = (c0 / sqrt((l * v))) * 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) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = (c0 / Math.sqrt((l * V))) * 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) <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = (c0 / math.sqrt((l * V))) * 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) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(Float64(c0 / sqrt(Float64(l * V))) * 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) <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = (c0 / sqrt((l * V))) * 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], 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < -0.0

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 71.9

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

   4. Applied rewrites 71.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 79.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. 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

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 91.1

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

   4. Applied rewrites 91.1%

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

Alternative 13: 73.7% 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 68.8%

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

Reproduce

herbie shell --seed 2025003 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))