Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.9% → 91.9%

Time: 15.1s

Alternatives: 14

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -4 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -4e-306)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (if (<= (* l V) 2e-315)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= (* l V) 1e+277)
       (* c0 (/ (sqrt A) (sqrt (* l V))))
       (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -4e-306) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else if ((l * V) <= 2e-315) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((l * V) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 * v) <= (-4d-306)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else if ((l * v) <= 2d-315) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((l * v) <= 1d+277) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -4e-306) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else if ((l * V) <= 2e-315) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((l * V) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -4e-306:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	elif (l * V) <= 2e-315:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (l * V) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -4e-306)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	elseif (Float64(l * V) <= 2e-315)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(l * V) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -4e-306)
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	elseif ((l * V) <= 2e-315)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((l * V) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -4e-306], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-315], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.00000000000000011e-306

   1. Initial program 69.9%

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

      1. associate-/r* 70.5%

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

      2. sqrt-div 49.5%

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

      3. div-inv 49.4%

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

   4. Applied egg-rr 49.4%

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

      1. associate-*r/ 49.5%

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

      2. *-rgt-identity 49.5%

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

   6. Simplified 49.5%

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

      1. frac-2neg 49.5%

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

      2. sqrt-div 53.9%

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

   8. Applied egg-rr 53.9%

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

   if -4.00000000000000011e-306 < (*.f64 V l) < 2.0000000019e-315

   1. Initial program 43.6%

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

      1. associate-/r* 66.6%

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

   3. Simplified 66.6%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 2.0000000019e-315 < (*.f64 V l) < 1e277

   1. Initial program 86.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 42.1%

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

      1. *-un-lft-identity 42.1%

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

      2. times-frac 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. frac-times 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 73.1%

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

      4. clear-num 73.1%

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

      5. associate-*l/ 73.1%

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

      6. *-un-lft-identity 73.1%

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

   6. Applied egg-rr 73.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -4 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 10^{-286} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (or (<= t_0 1e-286) (not (<= t_0 2e+299)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if ((t_0 <= 1e-286) || !(t_0 <= 2e+299)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if ((t_0 <= 1d-286) .or. (.not. (t_0 <= 2d+299))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if ((t_0 <= 1e-286) || !(t_0 <= 2e+299)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if (t_0 <= 1e-286) or not (t_0 <= 2e+299):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if ((t_0 <= 1e-286) || !(t_0 <= 2e+299))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if ((t_0 <= 1e-286) || ~((t_0 <= 2e+299)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 1e-286], N[Not[LessEqual[t$95$0, 2e+299]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000005e-286 or 2.0000000000000001e299 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.8%

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

      1. associate-/r* 70.1%

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

   3. Simplified 70.1%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 1.00000000000000005e-286 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e299

   1. Initial program 98.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-286} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (* (/ 1.0 V) (/ A l))))
     (if (<= t_0 5e+255) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else if (t_0 <= 5e+255) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else if (t_0 <= 5d+255) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else if (t_0 <= 5e+255) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	elif t_0 <= 5e+255:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	elseif (t_0 <= 5e+255)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	elseif (t_0 <= 5e+255)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+255], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.1%

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

      1. *-un-lft-identity 61.1%

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

      2. times-frac 67.3%

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

   4. Applied egg-rr 67.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000002e255

   1. Initial program 98.4%

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

   if 5.0000000000000002e255 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 62.9%

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

      1. associate-/r* 63.0%

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

      2. clear-num 63.0%

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

      3. sqrt-div 63.9%

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

      4. metadata-eval 63.9%

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

      5. div-inv 63.9%

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

      6. clear-num 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. associate-*r/ 65.1%

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

      2. sqrt-prod 53.5%

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

      3. times-frac 50.7%

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

      4. metadata-eval 50.7%

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

      5. sqrt-div 49.4%

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

      6. clear-num 49.4%

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

      7. associate-/r/ 52.1%

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

      8. sqrt-undiv 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. associate-/r/ 66.5%

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

      2. *-commutative 66.5%

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

   8. Simplified 66.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 10^{-286}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 1e-286)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+255) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 1e-286) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+255) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 1d-286) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+255) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 1e-286) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+255) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 1e-286:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+255:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if (t_0 <= 1e-286)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+255)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 1e-286)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+255)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-286], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+255], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000005e-286

   1. Initial program 61.1%

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

      1. associate-/r* 71.5%

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

   3. Simplified 71.5%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 1.00000000000000005e-286 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000002e255

   1. Initial program 98.4%

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

   if 5.0000000000000002e255 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 62.9%

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

      1. associate-/r* 63.0%

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

      2. clear-num 63.0%

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

      3. sqrt-div 63.9%

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

      4. metadata-eval 63.9%

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

      5. div-inv 63.9%

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

      6. clear-num 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. associate-*r/ 65.1%

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

      2. sqrt-prod 53.5%

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

      3. times-frac 50.7%

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

      4. metadata-eval 50.7%

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

      5. sqrt-div 49.4%

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

      6. clear-num 49.4%

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

      7. associate-/r/ 52.1%

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

      8. sqrt-undiv 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. associate-/r/ 66.5%

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

      2. *-commutative 66.5%

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

   8. Simplified 66.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-286}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-1}{\ell}}}{\frac{\sqrt{-V}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -2e-310)
   (* c0 (/ (sqrt (/ -1.0 l)) (/ (sqrt (- V)) (sqrt A))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-310) {
		tmp = c0 * (sqrt((-1.0 / l)) / (sqrt(-V) / sqrt(A)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 <= (-2d-310)) then
        tmp = c0 * (sqrt(((-1.0d0) / l)) / (sqrt(-v) / sqrt(a)))
    else
        tmp = c0 * (sqrt((a / v)) / 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 <= -2e-310) {
		tmp = c0 * (Math.sqrt((-1.0 / l)) / (Math.sqrt(-V) / Math.sqrt(A)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / 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 <= -2e-310:
		tmp = c0 * (math.sqrt((-1.0 / l)) / (math.sqrt(-V) / math.sqrt(A)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / 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 <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(-1.0 / l)) / Float64(sqrt(Float64(-V)) / sqrt(A))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / 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 <= -2e-310)
		tmp = c0 * (sqrt((-1.0 / l)) / (sqrt(-V) / sqrt(A)));
	else
		tmp = c0 * (sqrt((A / V)) / 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, -2e-310], N[(c0 * N[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 68.2%

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

      1. *-un-lft-identity 68.2%

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

      2. times-frac 69.6%

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

   4. Applied egg-rr 69.6%

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

      1. frac-times 68.2%

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

      2. *-commutative 68.2%

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

      3. frac-times 66.8%

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

      4. clear-num 66.5%

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

      5. associate-*l/ 66.5%

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

      6. *-un-lft-identity 66.5%

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

   6. Applied egg-rr 66.5%

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

      1. frac-2neg 66.5%

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

      2. sqrt-div 79.2%

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

      3. distribute-neg-frac 79.2%

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

      4. metadata-eval 79.2%

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

      5. distribute-neg-frac2 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. frac-2neg 79.2%

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

      2. sqrt-div 55.2%

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

      3. remove-double-neg 55.2%

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

   10. Applied egg-rr 55.2%

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

   if -1.999999999999994e-310 < l

   1. Initial program 71.3%

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

      1. associate-/r* 75.3%

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

      2. sqrt-div 90.3%

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

      3. div-inv 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.3%

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

      2. *-rgt-identity 90.3%

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

   6. Simplified 90.3%

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

Alternative 6: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -7.5 \cdot 10^{+264}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -7.5e+264)
   (/ (sqrt (/ A V)) (/ (sqrt l) c0))
   (if (<= (* l V) -4e-306)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* l V) 2e-315)
       (* c0 (sqrt (/ (/ A V) l)))
       (if (<= (* l V) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (sqrt (/ (/ 1.0 l) (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -7.5e+264) {
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	} else if ((l * V) <= -4e-306) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 2e-315) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((l * V) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 * v) <= (-7.5d+264)) then
        tmp = sqrt((a / v)) / (sqrt(l) / c0)
    else if ((l * v) <= (-4d-306)) then
        tmp = c0 * (sqrt(-a) / sqrt((l * -v)))
    else if ((l * v) <= 2d-315) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((l * v) <= 1d+277) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -7.5e+264) {
		tmp = Math.sqrt((A / V)) / (Math.sqrt(l) / c0);
	} else if ((l * V) <= -4e-306) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 2e-315) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((l * V) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -7.5e+264:
		tmp = math.sqrt((A / V)) / (math.sqrt(l) / c0)
	elif (l * V) <= -4e-306:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 2e-315:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (l * V) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -7.5e+264)
		tmp = Float64(sqrt(Float64(A / V)) / Float64(sqrt(l) / c0));
	elseif (Float64(l * V) <= -4e-306)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 2e-315)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(l * V) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -7.5e+264)
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	elseif ((l * V) <= -4e-306)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 2e-315)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((l * V) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -7.5e+264], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-306], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-315], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -7.50000000000000005e264

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. times-frac 63.9%

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

   4. Applied egg-rr 63.9%

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

      1. frac-times 41.6%

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

      2. *-commutative 41.6%

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

      3. frac-times 65.5%

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

      4. clear-num 65.5%

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

      5. associate-*l/ 65.5%

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

      6. *-un-lft-identity 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. div-inv 65.5%

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

      2. clear-num 65.5%

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

      3. sqrt-prod 57.2%

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

      4. inv-pow 57.2%

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

      5. sqrt-pow1 57.3%

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

      6. metadata-eval 57.3%

         \[\leadsto c0 \cdot \left({\ell}^{\color{blue}{-0.5}} \cdot \sqrt{\frac{A}{V}}\right) \]

      7. associate-*l* 57.3%

         \[\leadsto \color{blue}{\left(c0 \cdot {\ell}^{-0.5}\right) \cdot \sqrt{\frac{A}{V}}} \]

      8. *-commutative 57.3%

         \[\leadsto \color{blue}{\left({\ell}^{-0.5} \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

      9. metadata-eval 57.3%

         \[\leadsto \left({\ell}^{\color{blue}{\left(-0.5\right)}} \cdot c0\right) \cdot \sqrt{\frac{A}{V}} \]

      10. pow-flip 57.3%

          \[\leadsto \left(\color{blue}{\frac{1}{{\ell}^{0.5}}} \cdot c0\right) \cdot \sqrt{\frac{A}{V}} \]

      11. pow1/2 57.3%

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

      12. associate-/r/ 57.4%

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

      13. associate-*l/ 57.5%

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

      14. *-un-lft-identity 57.5%

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

   8. Applied egg-rr 57.5%

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

   if -7.50000000000000005e264 < (*.f64 V l) < -4.00000000000000011e-306

   1. Initial program 79.3%

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

      1. frac-2neg 79.3%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -4.00000000000000011e-306 < (*.f64 V l) < 2.0000000019e-315

   1. Initial program 43.6%

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

      1. associate-/r* 66.6%

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

   3. Simplified 66.6%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 2.0000000019e-315 < (*.f64 V l) < 1e277

   1. Initial program 86.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 42.1%

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

      1. *-un-lft-identity 42.1%

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

      2. times-frac 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. frac-times 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 73.1%

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

      4. clear-num 73.1%

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

      5. associate-*l/ 73.1%

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

      6. *-un-lft-identity 73.1%

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

   6. Applied egg-rr 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{\frac{V}{A}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -7.5 \cdot 10^{+264}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e+304)
   (* c0 (pow (* V (/ l A)) -0.5))
   (if (<= (* l V) -5e-128)
     (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
     (if (<= (* l V) 2e-315)
       (/ c0 (sqrt (/ l (/ A V))))
       (if (<= (* l V) 1e+277)
         (* c0 (/ (sqrt A) (sqrt (* l V))))
         (* c0 (sqrt (/ (/ 1.0 l) (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+304) {
		tmp = c0 * pow((V * (l / A)), -0.5);
	} else if ((l * V) <= -5e-128) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else if ((l * V) <= 2e-315) {
		tmp = c0 / sqrt((l / (A / V)));
	} else if ((l * V) <= 1e+277) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 * v) <= (-5d+304)) then
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    else if ((l * v) <= (-5d-128)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else if ((l * v) <= 2d-315) then
        tmp = c0 / sqrt((l / (a / v)))
    else if ((l * v) <= 1d+277) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+304) {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	} else if ((l * V) <= -5e-128) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((l * V) <= 2e-315) {
		tmp = c0 / Math.sqrt((l / (A / V)));
	} else if ((l * V) <= 1e+277) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e+304:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	elif (l * V) <= -5e-128:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	elif (l * V) <= 2e-315:
		tmp = c0 / math.sqrt((l / (A / V)))
	elif (l * V) <= 1e+277:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e+304)
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	elseif (Float64(l * V) <= -5e-128)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(l * V) <= 2e-315)
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	elseif (Float64(l * V) <= 1e+277)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e+304)
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	elseif ((l * V) <= -5e-128)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	elseif ((l * V) <= 2e-315)
		tmp = c0 / sqrt((l / (A / V)));
	elseif ((l * V) <= 1e+277)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e+304], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-128], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-315], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+277], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999997e304

   1. Initial program 33.3%

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

      1. associate-/r* 61.5%

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

      2. clear-num 61.5%

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

      3. sqrt-div 61.5%

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

      4. metadata-eval 61.5%

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

      5. div-inv 61.5%

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

      6. clear-num 61.5%

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

   4. Applied egg-rr 61.5%

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

      1. inv-pow 61.5%

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

      2. sqrt-prod 54.1%

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

      3. unpow-prod-down 54.1%

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

      4. clear-num 54.3%

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

      5. sqrt-div 54.1%

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

      6. metadata-eval 54.1%

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

      7. unpow-prod-down 54.1%

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

      8. div-inv 54.3%

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

      9. sqrt-undiv 61.5%

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

      10. sqrt-pow2 61.5%

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

      11. metadata-eval 61.5%

          \[\leadsto c0 \cdot {\left(\frac{\ell}{\frac{A}{V}}\right)}^{\color{blue}{-0.5}} \]

   6. Applied egg-rr 61.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{\ell}{\frac{A}{V}}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. associate-/r/ 61.7%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\ell}{A} \cdot V\right)}}^{-0.5} \]

      2. *-commutative 61.7%

         \[\leadsto c0 \cdot {\color{blue}{\left(V \cdot \frac{\ell}{A}\right)}}^{-0.5} \]

   8. Simplified 61.7%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]

   if -4.9999999999999997e304 < (*.f64 V l) < -5.0000000000000001e-128

   1. Initial program 83.1%

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

      1. clear-num 80.9%

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

      2. associate-/r/ 83.1%

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

      3. associate-/r* 83.2%

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

   4. Applied egg-rr 83.2%

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

   if -5.0000000000000001e-128 < (*.f64 V l) < 2.0000000019e-315

   1. Initial program 50.8%

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

      1. associate-/r* 66.6%

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

      2. clear-num 66.6%

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

      3. sqrt-div 67.1%

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

      4. metadata-eval 67.1%

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

      5. div-inv 67.1%

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

      6. clear-num 67.1%

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

   4. Applied egg-rr 67.1%

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

      1. associate-*r/ 67.2%

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

      2. sqrt-prod 42.7%

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

      3. times-frac 39.2%

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

      4. metadata-eval 39.2%

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

      5. sqrt-div 39.2%

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

      6. clear-num 39.3%

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

      7. associate-/r/ 42.6%

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

      8. sqrt-undiv 67.1%

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

   6. Applied egg-rr 67.1%

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

   if 2.0000000019e-315 < (*.f64 V l) < 1e277

   1. Initial program 86.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 1e277 < (*.f64 V l)

   1. Initial program 42.1%

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

      1. *-un-lft-identity 42.1%

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

      2. times-frac 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. frac-times 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 73.1%

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

      4. clear-num 73.1%

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

      5. associate-*l/ 73.1%

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

      6. *-un-lft-identity 73.1%

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

   6. Applied egg-rr 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{\frac{V}{A}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+277}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-163}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-1}{\ell}}}{\sqrt{\frac{V}{-A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -1.7e-163)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= l -2e-310)
     (* c0 (/ (sqrt (/ -1.0 l)) (sqrt (/ V (- A)))))
     (* c0 (/ (sqrt (/ A V)) (sqrt l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -1.7e-163) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if (l <= -2e-310) {
		tmp = c0 * (sqrt((-1.0 / l)) / sqrt((V / -A)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 <= (-1.7d-163)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if (l <= (-2d-310)) then
        tmp = c0 * (sqrt(((-1.0d0) / l)) / sqrt((v / -a)))
    else
        tmp = c0 * (sqrt((a / v)) / 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 <= -1.7e-163) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if (l <= -2e-310) {
		tmp = c0 * (Math.sqrt((-1.0 / l)) / Math.sqrt((V / -A)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / 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 <= -1.7e-163:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif l <= -2e-310:
		tmp = c0 * (math.sqrt((-1.0 / l)) / math.sqrt((V / -A)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / 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 <= -1.7e-163)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (l <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(-1.0 / l)) / sqrt(Float64(V / Float64(-A)))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / 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 <= -1.7e-163)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif (l <= -2e-310)
		tmp = c0 * (sqrt((-1.0 / l)) / sqrt((V / -A)));
	else
		tmp = c0 * (sqrt((A / V)) / 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, -1.7e-163], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(c0 * N[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(V / (-A)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.70000000000000007e-163

   1. Initial program 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. frac-times 71.1%

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

      2. *-commutative 71.1%

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

      3. frac-times 67.8%

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

      4. clear-num 67.5%

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

      5. associate-*l/ 67.5%

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

      6. *-un-lft-identity 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. associate-/l/ 65.9%

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

      2. sqrt-div 66.1%

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

      3. associate-/r/ 72.7%

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

      4. sqrt-div 72.5%

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

      5. clear-num 73.2%

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

      6. frac-2neg 73.2%

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

      7. sqrt-div 51.9%

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

      8. distribute-neg-frac2 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. distribute-frac-neg2 51.9%

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

      2. distribute-neg-frac 51.9%

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

   10. Simplified 51.9%

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

   if -1.70000000000000007e-163 < l < -1.999999999999994e-310

   1. Initial program 58.4%

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

      1. *-un-lft-identity 58.4%

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

      2. times-frac 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. frac-times 58.4%

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

      2. *-commutative 58.4%

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

      3. frac-times 63.4%

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

      4. clear-num 63.4%

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

      5. associate-*l/ 63.4%

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

      6. *-un-lft-identity 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. frac-2neg 63.4%

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

      2. sqrt-div 84.7%

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

      3. distribute-neg-frac 84.7%

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

      4. metadata-eval 84.7%

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

      5. distribute-neg-frac2 84.7%

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

   8. Applied egg-rr 84.7%

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

   if -1.999999999999994e-310 < l

   1. Initial program 71.3%

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

      1. associate-/r* 75.3%

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

      2. sqrt-div 90.3%

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

      3. div-inv 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.3%

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

      2. *-rgt-identity 90.3%

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

   6. Simplified 90.3%

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

Alternative 9: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-162}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-V}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -1.4e-162)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= l -2e-310)
     (* c0 (/ (sqrt (/ A (- V))) (sqrt (- l))))
     (* c0 (/ (sqrt (/ A V)) (sqrt l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -1.4e-162) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if (l <= -2e-310) {
		tmp = c0 * (sqrt((A / -V)) / sqrt(-l));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 <= (-1.4d-162)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if (l <= (-2d-310)) then
        tmp = c0 * (sqrt((a / -v)) / sqrt(-l))
    else
        tmp = c0 * (sqrt((a / v)) / 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 <= -1.4e-162) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if (l <= -2e-310) {
		tmp = c0 * (Math.sqrt((A / -V)) / Math.sqrt(-l));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / 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 <= -1.4e-162:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif l <= -2e-310:
		tmp = c0 * (math.sqrt((A / -V)) / math.sqrt(-l))
	else:
		tmp = c0 * (math.sqrt((A / V)) / 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 <= -1.4e-162)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (l <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-V))) / sqrt(Float64(-l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / 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 <= -1.4e-162)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif (l <= -2e-310)
		tmp = c0 * (sqrt((A / -V)) / sqrt(-l));
	else
		tmp = c0 * (sqrt((A / V)) / 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, -1.4e-162], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(c0 * N[(N[Sqrt[N[(A / (-V)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.40000000000000011e-162

   1. Initial program 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. times-frac 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. frac-times 71.1%

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

      2. *-commutative 71.1%

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

      3. frac-times 67.8%

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

      4. clear-num 67.5%

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

      5. associate-*l/ 67.5%

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

      6. *-un-lft-identity 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. associate-/l/ 65.9%

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

      2. sqrt-div 66.1%

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

      3. associate-/r/ 72.7%

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

      4. sqrt-div 72.5%

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

      5. clear-num 73.2%

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

      6. frac-2neg 73.2%

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

      7. sqrt-div 51.9%

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

      8. distribute-neg-frac2 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. distribute-frac-neg2 51.9%

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

      2. distribute-neg-frac 51.9%

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

   10. Simplified 51.9%

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

   if -1.40000000000000011e-162 < l < -1.999999999999994e-310

   1. Initial program 58.4%

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

      1. associate-/r* 63.3%

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

      2. frac-2neg 63.3%

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

      3. sqrt-div 84.5%

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

      4. distribute-neg-frac2 84.5%

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

   4. Applied egg-rr 84.5%

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

   if -1.999999999999994e-310 < l

   1. Initial program 71.3%

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

      1. associate-/r* 75.3%

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

      2. sqrt-div 90.3%

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

      3. div-inv 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.3%

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

      2. *-rgt-identity 90.3%

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

   6. Simplified 90.3%

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

Alternative 10: 83.7% accurate, 0.5× speedup

Math

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

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 -2e-310)
   (* c0 (/ (sqrt A) (sqrt (* l V))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 <= (-2d-310)) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((a / v)) / 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 <= -2e-310) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / 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 <= -2e-310:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / 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 <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / 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 <= -2e-310)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 * (sqrt((A / V)) / 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, -2e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 68.2%

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

      1. sqrt-div 43.9%

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

      2. div-inv 43.9%

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

   4. Applied egg-rr 43.9%

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

      1. associate-*r/ 43.9%

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

      2. *-rgt-identity 43.9%

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

   6. Simplified 43.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 71.3%

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

      1. associate-/r* 75.3%

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

      2. sqrt-div 90.3%

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

      3. div-inv 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.3%

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

      2. *-rgt-identity 90.3%

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

   6. Simplified 90.3%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-323} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+212}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \end{array} \end{array} \]

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 (/ A (* l V))))
   (if (or (<= t_0 2e-323) (not (<= t_0 2e+212)))
     (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))
     (* c0 (sqrt (* A (/ (/ 1.0 V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if ((t_0 <= 2e-323) || !(t_0 <= 2e+212)) {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	} else {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 / (l * v)
    if ((t_0 <= 2d-323) .or. (.not. (t_0 <= 2d+212))) then
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    else
        tmp = c0 * sqrt((a * ((1.0d0 / 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 = A / (l * V);
	double tmp;
	if ((t_0 <= 2e-323) || !(t_0 <= 2e+212)) {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	} else {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if (t_0 <= 2e-323) or not (t_0 <= 2e+212):
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	else:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if ((t_0 <= 2e-323) || !(t_0 <= 2e+212))
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	else
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / 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 / (l * V);
	tmp = 0.0;
	if ((t_0 <= 2e-323) || ~((t_0 <= 2e+212)))
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	else
		tmp = c0 * sqrt((A * ((1.0 / 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[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-323], N[Not[LessEqual[t$95$0, 2e+212]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 1.97626e-323 or 1.9999999999999998e212 < (/.f64 A (*.f64 V l))

   1. Initial program 41.2%

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

      1. *-un-lft-identity 41.2%

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

      2. times-frac 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. frac-times 41.2%

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

      2. *-commutative 41.2%

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

      3. frac-times 55.6%

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

      4. clear-num 55.6%

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

      5. associate-*l/ 56.1%

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

      6. *-un-lft-identity 56.1%

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

   6. Applied egg-rr 56.1%

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

   if 1.97626e-323 < (/.f64 A (*.f64 V l)) < 1.9999999999999998e212

   1. Initial program 99.1%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 99.1%

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

      3. associate-/r* 99.2%

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{-323} \lor \neg \left(\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+212}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-323}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+212}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 2e-323)
     (* c0 (sqrt (/ (/ 1.0 l) (/ V A))))
     (if (<= t_0 2e+212)
       (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
       (* c0 (/ 1.0 (sqrt (* l (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-323) {
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	} else if (t_0 <= 2e+212) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 / (l * v)
    if (t_0 <= 2d-323) then
        tmp = c0 * sqrt(((1.0d0 / l) / (v / a)))
    else if (t_0 <= 2d+212) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else
        tmp = c0 * (1.0d0 / sqrt((l * (v / a))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-323) {
		tmp = c0 * Math.sqrt(((1.0 / l) / (V / A)));
	} else if (t_0 <= 2e+212) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 2e-323:
		tmp = c0 * math.sqrt(((1.0 / l) / (V / A)))
	elif t_0 <= 2e+212:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 2e-323)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / l) / Float64(V / A))));
	elseif (t_0 <= 2e+212)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(l * Float64(V / A)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 2e-323)
		tmp = c0 * sqrt(((1.0 / l) / (V / A)));
	elseif (t_0 <= 2e+212)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	else
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-323], N[(c0 * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+212], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.97626e-323

   1. Initial program 35.0%

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

      1. *-un-lft-identity 35.0%

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

      2. times-frac 55.2%

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

   4. Applied egg-rr 55.2%

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

      1. frac-times 35.0%

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

      2. *-commutative 35.0%

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

      3. frac-times 55.4%

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

      4. clear-num 55.4%

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

      5. associate-*l/ 55.4%

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

      6. *-un-lft-identity 55.4%

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

   6. Applied egg-rr 55.4%

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

   if 1.97626e-323 < (/.f64 A (*.f64 V l)) < 1.9999999999999998e212

   1. Initial program 99.1%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 99.1%

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

      3. associate-/r* 99.2%

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

   4. Applied egg-rr 99.2%

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

   if 1.9999999999999998e212 < (/.f64 A (*.f64 V l))

   1. Initial program 47.1%

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

      1. associate-/r* 55.7%

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

      2. clear-num 55.7%

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

      3. sqrt-div 56.5%

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

      4. metadata-eval 56.5%

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

      5. div-inv 56.5%

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

      6. clear-num 57.6%

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

   4. Applied egg-rr 57.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{-323}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{\ell}}{\frac{V}{A}}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+212}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+212}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (* (/ 1.0 V) (/ A l))))
     (if (<= t_0 2e+212)
       (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else if (t_0 <= 2e+212) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    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 / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else if (t_0 <= 2d+212) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else if (t_0 <= 2e+212) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	elif t_0 <= 2e+212:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	elseif (t_0 <= 2e+212)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	elseif (t_0 <= 2e+212)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+212], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

      1. *-un-lft-identity 35.2%

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

      2. times-frac 55.9%

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

   4. Applied egg-rr 55.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.9999999999999998e212

   1. Initial program 98.5%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 98.5%

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

      3. associate-/r* 98.6%

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

   4. Applied egg-rr 98.6%

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

   if 1.9999999999999998e212 < (/.f64 A (*.f64 V l))

   1. Initial program 47.1%

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

      1. associate-/r* 55.7%

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

      2. clear-num 55.7%

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

      3. sqrt-div 56.5%

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

      4. metadata-eval 56.5%

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

      5. div-inv 56.5%

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

      6. clear-num 57.6%

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

   4. Applied egg-rr 57.6%

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

      1. associate-*r/ 57.7%

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

      2. sqrt-prod 39.3%

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

      3. times-frac 39.2%

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

      4. metadata-eval 39.2%

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

      5. sqrt-div 38.2%

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

      6. clear-num 38.2%

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

      7. associate-/r/ 38.2%

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

      8. sqrt-undiv 56.5%

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

   6. Applied egg-rr 56.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+212}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \end{array} \]

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 (* l V)))))

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (l * v)))
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 / (l * V)));
}

Python

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

Julia

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

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (l * V)));
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[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

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

3. Final simplification 69.7%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))