Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.3% → 92.0%

Time: 6.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% 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: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := {\left(-A\right)}^{0.25}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \left(\frac{t\_0}{\sqrt{-V}} \cdot \frac{t\_0}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (pow (- A) 0.25)))
   (if (<= (* V l) -5e-314)
     (* c0 (* (/ t_0 (sqrt (- V))) (/ t_0 (sqrt l))))
     (if (<= (* V l) 0.0)
       (/ c0 (sqrt (* (/ V A) l)))
       (if (<= (* V l) 2e+276)
         (/ c0 (* (pow A -0.5) (sqrt (* l V))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = pow(-A, 0.25);
	double tmp;
	if ((V * l) <= -5e-314) {
		tmp = c0 * ((t_0 / sqrt(-V)) * (t_0 / sqrt(l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / sqrt(((V / A) * l));
	} else if ((V * l) <= 2e+276) {
		tmp = c0 / (pow(A, -0.5) * sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a ** 0.25d0
    if ((v * l) <= (-5d-314)) then
        tmp = c0 * ((t_0 / sqrt(-v)) * (t_0 / sqrt(l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / sqrt(((v / a) * l))
    else if ((v * l) <= 2d+276) then
        tmp = c0 / ((a ** (-0.5d0)) * sqrt((l * v)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.pow(-A, 0.25);
	double tmp;
	if ((V * l) <= -5e-314) {
		tmp = c0 * ((t_0 / Math.sqrt(-V)) * (t_0 / Math.sqrt(l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else if ((V * l) <= 2e+276) {
		tmp = c0 / (Math.pow(A, -0.5) * Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.pow(-A, 0.25)
	tmp = 0
	if (V * l) <= -5e-314:
		tmp = c0 * ((t_0 / math.sqrt(-V)) * (t_0 / math.sqrt(l)))
	elif (V * l) <= 0.0:
		tmp = c0 / math.sqrt(((V / A) * l))
	elif (V * l) <= 2e+276:
		tmp = c0 / (math.pow(A, -0.5) * math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(-A) ^ 0.25
	tmp = 0.0
	if (Float64(V * l) <= -5e-314)
		tmp = Float64(c0 * Float64(Float64(t_0 / sqrt(Float64(-V))) * Float64(t_0 / sqrt(l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	elseif (Float64(V * l) <= 2e+276)
		tmp = Float64(c0 / Float64((A ^ -0.5) * sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = -A ^ 0.25;
	tmp = 0.0;
	if ((V * l) <= -5e-314)
		tmp = c0 * ((t_0 / sqrt(-V)) * (t_0 / sqrt(l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / sqrt(((V / A) * l));
	elseif ((V * l) <= 2e+276)
		tmp = c0 / ((A ^ -0.5) * sqrt((l * V)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Power[(-A), 0.25], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e-314], N[(c0 * N[(N[(t$95$0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+276], N[(c0 / N[(N[Power[A, -0.5], $MachinePrecision] * N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.99999999982e-314

   1. Initial program 76.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 48.8%

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

   if -4.99999999982e-314 < (*.f64 V l) < -0.0

   1. Initial program 41.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 41.5%

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

   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 41.5

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

   7. Applied rewrites 41.5%

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

   9. Applied rewrites 69.1%

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

       1. lift-*.f64 N/A

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

       2. *-rgt-identity 69.1

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

   11. Applied rewrites 69.1%

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

   if -0.0 < (*.f64 V l) < 2.0000000000000001e276

   1. Initial program 86.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 86.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. sqrt-pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 99.1

          \[\leadsto \frac{c0 \cdot 1}{{A}^{-0.5} \cdot \color{blue}{\sqrt{\ell \cdot V}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{c0 \cdot 1}{\color{blue}{{A}^{-0.5} \cdot \sqrt{\ell \cdot V}}} \]

   if 2.0000000000000001e276 < (*.f64 V l)

   1. Initial program 48.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.7

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

   4. Applied rewrites 75.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \left(\frac{{\left(-A\right)}^{0.25}}{\sqrt{-V}} \cdot \frac{{\left(-A\right)}^{0.25}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0}{{A}^{-0.5} \cdot \sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-162} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+91}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot t\_0\\ \end{array} \end{array} \]

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 (sqrt (/ A (* V l)))))
   (if (or (<= t_0 5e-162) (not (<= t_0 5e+91)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 t_0))))

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-162) || !(t_0 <= 5e+91)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * 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 = sqrt((a / (v * l)))
    if ((t_0 <= 5d-162) .or. (.not. (t_0 <= 5d+91))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * 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 = Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 5e-162) || !(t_0 <= 5e+91)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * t_0;
	}
	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-162) or not (t_0 <= 5e+91):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * t_0
	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-162) || !(t_0 <= 5e+91))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * 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 = sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 5e-162) || ~((t_0 <= 5e+91)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * 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[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-162], N[Not[LessEqual[t$95$0, 5e+91]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.00000000000000014e-162 or 5.0000000000000002e91 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 42.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. 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 57.2

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

   4. Applied rewrites 57.2%

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

   if 5.00000000000000014e-162 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.0000000000000002e91

   1. Initial program 99.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-162} \lor \neg \left(\sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+91}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;c0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

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 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+114) (* c0 t_0) (/ 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 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+114) {
		tmp = c0 * t_0;
	} 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 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+114) then
        tmp = c0 * t_0
    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 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+114) {
		tmp = c0 * t_0;
	} 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 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+114:
		tmp = c0 * t_0
	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 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+114)
		tmp = Float64(c0 * t_0);
	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 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+114)
		tmp = c0 * t_0;
	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, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+114], N[(c0 * t$95$0), $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))) < 0.0

   1. Initial program 31.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 52.9

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

   4. Applied rewrites 52.9%

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

   if 0.0 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.0000000000000001e114

   1. Initial program 99.1%

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

   if 5.0000000000000001e114 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 48.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 48.3%

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

   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 51.6

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

   7. Applied rewrites 51.6%

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

   9. Applied rewrites 63.0%

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

       1. lift-*.f64 N/A

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

       2. *-rgt-identity 63.0

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

   11. Applied rewrites 63.0%

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

Alternative 4: 78.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;c0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A (* V l)))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+91) (* c0 t_0) (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+91) {
		tmp = c0 * t_0;
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+91) then
        tmp = c0 * t_0
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+91) {
		tmp = c0 * t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+91:
		tmp = c0 * t_0
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / Float64(V * l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+91)
		tmp = Float64(c0 * t_0);
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+91)
		tmp = c0 * t_0;
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+91], N[(c0 * t$95$0), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 52.9

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

   4. Applied rewrites 52.9%

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

   if 0.0 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.0000000000000002e91

   1. Initial program 99.0%

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

   if 5.0000000000000002e91 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 52.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 61.8

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

   4. Applied rewrites 61.8%

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

Alternative 5: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-127}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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) -5e-127)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 0.0)
     (/ c0 (sqrt (* (/ l A) V)))
     (if (<= (* V l) 1e+302)
       (* c0 (/ (sqrt A) (sqrt (* l V))))
       (* c0 (sqrt (/ (/ A 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) <= -5e-127) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / sqrt(((l / A) * V));
	} else if ((V * l) <= 1e+302) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((A / 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) <= (-5d-127)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / sqrt(((l / a) * v))
    else if ((v * l) <= 1d+302) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((a / 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) <= -5e-127) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else if ((V * l) <= 1e+302) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((A / 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) <= -5e-127:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 0.0:
		tmp = c0 / math.sqrt(((l / A) * V))
	elif (V * l) <= 1e+302:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((A / 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) <= -5e-127)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	elseif (Float64(V * l) <= 1e+302)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / 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) <= -5e-127)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 0.0)
		tmp = c0 / sqrt(((l / A) * V));
	elseif ((V * l) <= 1e+302)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((A / 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], -5e-127], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+302], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999997e-127

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 41.8

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

   4. Applied rewrites 41.8%

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

   if -4.9999999999999997e-127 < (*.f64 V l) < -0.0

   1. Initial program 64.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. sqrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 43.9

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

   4. Applied rewrites 43.9%

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

   6. Applied rewrites 77.1%

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

   if -0.0 < (*.f64 V l) < 1.0000000000000001e302

   1. Initial program 86.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 1.0000000000000001e302 < (*.f64 V l)

   1. Initial program 44.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 6: 82.2% accurate, 0.5× speedup

Math

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

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) l)))
   (if (<= (* V l) 1e+302)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (* c0 (sqrt (/ (/ A 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) * l));
	} else if ((V * l) <= 1e+302) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((A / 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) * l))
    else if ((v * l) <= 1d+302) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((a / 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) * l));
	} else if ((V * l) <= 1e+302) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((A / 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) * l))
	elif (V * l) <= 1e+302:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((A / 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(V / A) * l)));
	elseif (Float64(V * l) <= 1e+302)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / 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) * l));
	elseif ((V * l) <= 1e+302)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((A / 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[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+302], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 69.8%

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

   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

   9. Applied rewrites 73.6%

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

       1. lift-*.f64 N/A

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

       2. *-rgt-identity 73.6

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

   11. Applied rewrites 73.6%

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

   if -0.0 < (*.f64 V l) < 1.0000000000000001e302

   1. Initial program 86.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 1.0000000000000001e302 < (*.f64 V l)

   1. Initial program 44.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 7: 88.1% accurate, 0.5× speedup

Math

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

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

   1. Initial program 74.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 45.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-sqrt.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

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

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 53.9

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

   6. Applied rewrites 53.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.2%

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

   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 73.0

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

   7. Applied rewrites 73.0%

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

   9. Applied rewrites 86.8%

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

4. Final simplification 70.2%

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

Alternative 8: 74.3% 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 73.6%

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

Reproduce

herbie shell --seed 2025006 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))