Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 89.9%

Time: 26.0s

Alternatives: 12

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: 73.5% 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: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \frac{\sqrt{-A}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V} \cdot \sqrt{\frac{1}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* (/ c0 (sqrt l)) (/ (sqrt (- A)) (sqrt (- V))))
   (/ c0 (* (sqrt (* l V)) (sqrt (/ 1.0 A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (c0 / sqrt(l)) * (sqrt(-A) / sqrt(-V));
	} else {
		tmp = c0 / (sqrt((l * V)) * sqrt((1.0 / 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 (a <= (-5d-310)) then
        tmp = (c0 / sqrt(l)) * (sqrt(-a) / sqrt(-v))
    else
        tmp = c0 / (sqrt((l * v)) * sqrt((1.0d0 / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (c0 / Math.sqrt(l)) * (Math.sqrt(-A) / Math.sqrt(-V));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) * Math.sqrt((1.0 / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = (c0 / math.sqrt(l)) * (math.sqrt(-A) / math.sqrt(-V))
	else:
		tmp = c0 / (math.sqrt((l * V)) * math.sqrt((1.0 / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(Float64(c0 / sqrt(l)) * Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) * sqrt(Float64(1.0 / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = (c0 / sqrt(l)) * (sqrt(-A) / sqrt(-V));
	else
		tmp = c0 / (sqrt((l * V)) * sqrt((1.0 / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-/r* 75.0%

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

      3. sqrt-div 46.2%

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

      4. associate-*l/ 44.7%

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

   4. Applied egg-rr 44.7%

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

      1. associate-/l* 46.1%

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

      2. *-commutative 46.1%

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

   6. Applied egg-rr 46.1%

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

      1. frac-2neg 46.1%

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

      2. sqrt-div 55.3%

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

   8. Applied egg-rr 55.3%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. times-frac 75.8%

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

   4. Applied egg-rr 75.8%

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

      1. *-commutative 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*r/ 76.0%

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

      4. frac-2neg 76.0%

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

      5. associate-*l/ 76.0%

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

      6. *-un-lft-identity 76.0%

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

      7. distribute-frac-neg2 76.0%

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

      8. sqrt-undiv 41.6%

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

      9. clear-num 41.6%

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

      10. un-div-inv 41.6%

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

      11. sqrt-undiv 76.1%

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

      12. add-sqr-sqrt 38.8%

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

      13. sqrt-unprod 24.0%

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

      14. sqr-neg 24.0%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.0%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 23.4%

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

   6. Applied egg-rr 76.1%

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

      1. associate-/r/ 75.5%

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

      2. *-commutative 75.5%

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

   8. Simplified 75.5%

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

      1. pow1/2 75.5%

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

      2. associate-*r/ 76.7%

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

      3. div-inv 76.6%

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

      4. unpow-prod-down 85.4%

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

      5. pow1/2 85.4%

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

   10. Applied egg-rr 85.4%

       \[\leadsto \frac{c0}{\color{blue}{\sqrt{V \cdot \ell} \cdot {\left(\frac{1}{A}\right)}^{0.5}}} \]
   11. Step-by-step derivation

       1. unpow1/2 85.4%

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

   12. Simplified 85.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \frac{\sqrt{-A}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V} \cdot \sqrt{\frac{1}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+193}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 0.0) (not (<= t_0 5e+193)))
     (* 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 <= 0.0) || !(t_0 <= 5e+193)) {
		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 <= 0.0d0) .or. (.not. (t_0 <= 5d+193))) 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 <= 0.0) || !(t_0 <= 5e+193)) {
		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 <= 0.0) or not (t_0 <= 5e+193):
		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 <= 0.0) || !(t_0 <= 5e+193))
		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 <= 0.0) || ~((t_0 <= 5e+193)))
		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, 0.0], N[Not[LessEqual[t$95$0, 5e+193]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.99999999999999972e193 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.1%

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

      1. associate-/r* 72.5%

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

   3. Simplified 72.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.99999999999999972e193

   1. Initial program 99.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 5 \cdot 10^{+193}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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)
     (sqrt (* (/ A l) (* c0 (/ c0 V))))
     (if (<= t_0 4e+232) 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 = sqrt(((A / l) * (c0 * (c0 / V))));
	} else if (t_0 <= 4e+232) {
		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 = sqrt(((a / l) * (c0 * (c0 / v))))
    else if (t_0 <= 4d+232) 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 = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else if (t_0 <= 4e+232) {
		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 = math.sqrt(((A / l) * (c0 * (c0 / V))))
	elif t_0 <= 4e+232:
		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 = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	elseif (t_0 <= 4e+232)
		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 = sqrt(((A / l) * (c0 * (c0 / V))));
	elseif (t_0 <= 4e+232)
		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[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+232], 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 67.4%

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

      1. add-sqr-sqrt 14.4%

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

      2. sqrt-unprod 15.9%

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

      3. *-commutative 15.9%

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

      4. *-commutative 15.9%

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

      5. swap-sqr 15.6%

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

      6. add-sqr-sqrt 15.6%

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

      7. pow2 15.6%

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

   4. Applied egg-rr 15.6%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 16.3%

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

      2. *-commutative 16.3%

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

      3. times-frac 18.7%

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

   6. Simplified 18.7%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{\ell} \cdot \frac{{c0}^{2}}{V}}} \]
   7. Step-by-step derivation

      1. unpow2 18.7%

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

      2. associate-/l* 20.0%

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

   8. Applied egg-rr 20.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.00000000000000023e232

   1. Initial program 99.6%

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

   if 4.00000000000000023e232 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 70.8%

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

   4. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*r/ 70.8%

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

      4. frac-2neg 70.8%

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

      5. associate-*l/ 70.8%

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

      6. *-un-lft-identity 70.8%

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

      7. distribute-frac-neg2 70.8%

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

      8. sqrt-undiv 36.5%

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

      9. clear-num 36.5%

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

      10. un-div-inv 36.6%

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

      11. sqrt-undiv 70.9%

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

      12. add-sqr-sqrt 27.8%

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

      13. sqrt-unprod 16.2%

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

      14. sqr-neg 16.2%

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

      15. sqrt-unprod 0.1%

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

      16. add-sqr-sqrt 0.1%

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

      17. add-sqr-sqrt 0.1%

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

      18. sqrt-unprod 16.1%

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

   6. Applied egg-rr 70.9%

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

      1. associate-/r/ 70.8%

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

      2. *-commutative 70.8%

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

   8. Simplified 70.8%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 4 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 (/ (/ A V) l)))
     (if (<= t_0 4e+232) 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(((A / V) / l));
	} else if (t_0 <= 4e+232) {
		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(((a / v) / l))
    else if (t_0 <= 4d+232) 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(((A / V) / l));
	} else if (t_0 <= 4e+232) {
		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(((A / V) / l))
	elif t_0 <= 4e+232:
		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(A / V) / l)));
	elseif (t_0 <= 4e+232)
		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(((A / V) / l));
	elseif (t_0 <= 4e+232)
		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[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+232], 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 67.4%

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

      1. associate-/r* 72.4%

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

   3. Simplified 72.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.00000000000000023e232

   1. Initial program 99.6%

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

   if 4.00000000000000023e232 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 70.8%

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

   4. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*r/ 70.8%

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

      4. frac-2neg 70.8%

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

      5. associate-*l/ 70.8%

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

      6. *-un-lft-identity 70.8%

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

      7. distribute-frac-neg2 70.8%

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

      8. sqrt-undiv 36.5%

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

      9. clear-num 36.5%

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

      10. un-div-inv 36.6%

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

      11. sqrt-undiv 70.9%

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

      12. add-sqr-sqrt 27.8%

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

      13. sqrt-unprod 16.2%

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

      14. sqr-neg 16.2%

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

      15. sqrt-unprod 0.1%

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

      16. add-sqr-sqrt 0.1%

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

      17. add-sqr-sqrt 0.1%

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

      18. sqrt-unprod 16.1%

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

   6. Applied egg-rr 70.9%

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

      1. associate-/r/ 70.8%

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

      2. *-commutative 70.8%

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

   8. Simplified 70.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 4 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot \left(-V\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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) -2e+282)
   (/ (* c0 (sqrt (/ A V))) (sqrt l))
   (if (<= (* l V) -1e-266)
     (/ c0 (/ (sqrt (* l (- V))) (sqrt (- A))))
     (if (<= (* l V) 2e-312)
       (/ c0 (sqrt (* l (/ V A))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e+282) {
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	} else if ((l * V) <= -1e-266) {
		tmp = c0 / (sqrt((l * -V)) / sqrt(-A));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((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) :: tmp
    if ((l * v) <= (-2d+282)) then
        tmp = (c0 * sqrt((a / v))) / sqrt(l)
    else if ((l * v) <= (-1d-266)) then
        tmp = c0 / (sqrt((l * -v)) / sqrt(-a))
    else if ((l * v) <= 2d-312) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((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 tmp;
	if ((l * V) <= -2e+282) {
		tmp = (c0 * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((l * V) <= -1e-266) {
		tmp = c0 / (Math.sqrt((l * -V)) / Math.sqrt(-A));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -2e+282:
		tmp = (c0 * math.sqrt((A / V))) / math.sqrt(l)
	elif (l * V) <= -1e-266:
		tmp = c0 / (math.sqrt((l * -V)) / math.sqrt(-A))
	elif (l * V) <= 2e-312:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -2e+282)
		tmp = Float64(Float64(c0 * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(l * V) <= -1e-266)
		tmp = Float64(c0 / Float64(sqrt(Float64(l * Float64(-V))) / sqrt(Float64(-A))));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(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)
	tmp = 0.0;
	if ((l * V) <= -2e+282)
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	elseif ((l * V) <= -1e-266)
		tmp = c0 / (sqrt((l * -V)) / sqrt(-A));
	elseif ((l * V) <= 2e-312)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((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_] := If[LessEqual[N[(l * V), $MachinePrecision], -2e+282], N[(N[(c0 * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-266], N[(c0 / N[(N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000007e282

   1. Initial program 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-/r* 73.1%

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

      3. sqrt-div 35.5%

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

      4. associate-*l/ 35.5%

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

   4. Applied egg-rr 35.5%

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

   if -2.00000000000000007e282 < (*.f64 V l) < -9.9999999999999998e-267

   1. Initial program 79.9%

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

      1. *-un-lft-identity 79.9%

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

      2. times-frac 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. *-commutative 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*r/ 76.2%

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

      4. frac-2neg 76.2%

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

      5. associate-*l/ 76.1%

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

      6. *-un-lft-identity 76.1%

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

      7. distribute-frac-neg2 76.1%

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

      8. sqrt-undiv 43.4%

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

      9. clear-num 43.4%

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

      10. un-div-inv 43.5%

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

      11. sqrt-undiv 75.0%

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

      12. add-sqr-sqrt 32.8%

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

      13. sqrt-unprod 23.9%

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

      14. sqr-neg 23.9%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.0%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 20.6%

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

   6. Applied egg-rr 75.0%

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

      1. associate-/r/ 67.9%

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

      2. *-commutative 67.9%

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

   8. Simplified 67.9%

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

      1. associate-*r/ 79.6%

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

      2. frac-2neg 79.6%

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

      3. sqrt-div 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. distribute-rgt-neg-out 99.4%

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

       2. *-commutative 99.4%

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

       3. distribute-rgt-neg-in 99.4%

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

   12. Simplified 99.4%

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

   if -9.9999999999999998e-267 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 50.4%

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

      1. *-un-lft-identity 50.4%

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

      2. times-frac 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*r/ 75.0%

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

      4. frac-2neg 75.0%

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

      5. associate-*l/ 75.0%

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

      6. *-un-lft-identity 75.0%

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

      7. distribute-frac-neg2 75.0%

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

      8. sqrt-undiv 33.3%

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

      9. clear-num 33.3%

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

      10. un-div-inv 33.4%

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

      11. sqrt-undiv 75.1%

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

      12. add-sqr-sqrt 26.5%

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

      13. sqrt-unprod 17.6%

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

      14. sqr-neg 17.6%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.1%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 18.3%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/r/ 75.0%

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

      2. associate-*l/ 50.5%

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

      3. associate-*r/ 75.1%

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

   8. Simplified 75.1%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. pow1/2 82.2%

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

      2. div-inv 82.1%

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

      3. unpow-prod-down 92.2%

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

      4. pow1/2 92.2%

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

   4. Applied egg-rr 92.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(\frac{1}{V \cdot \ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 92.2%

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

      2. associate-/r* 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot \left(-V\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-266}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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) (- INFINITY))
   (/ (* c0 (sqrt (/ A V))) (sqrt l))
   (if (<= (* l V) -1e-266)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* l V) 2e-312)
       (/ c0 (sqrt (* l (/ V A))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	} else if ((l * V) <= -1e-266) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((l * V) <= -1e-266) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (c0 * math.sqrt((A / V))) / math.sqrt(l)
	elif (l * V) <= -1e-266:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 2e-312:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(l * V) <= -1e-266)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(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)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	elseif ((l * V) <= -1e-266)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 2e-312)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((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_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-266], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/r* 68.5%

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

      3. sqrt-div 35.8%

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

      4. associate-*l/ 35.8%

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

   4. Applied egg-rr 35.8%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999998e-267

   1. Initial program 80.6%

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

      1. frac-2neg 80.6%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-267 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 50.4%

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

      1. *-un-lft-identity 50.4%

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

      2. times-frac 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*r/ 75.0%

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

      4. frac-2neg 75.0%

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

      5. associate-*l/ 75.0%

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

      6. *-un-lft-identity 75.0%

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

      7. distribute-frac-neg2 75.0%

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

      8. sqrt-undiv 33.3%

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

      9. clear-num 33.3%

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

      10. un-div-inv 33.4%

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

      11. sqrt-undiv 75.1%

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

      12. add-sqr-sqrt 26.5%

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

      13. sqrt-unprod 17.6%

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

      14. sqr-neg 17.6%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.1%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 18.3%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/r/ 75.0%

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

      2. associate-*l/ 50.5%

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

      3. associate-*r/ 75.1%

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

   8. Simplified 75.1%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. pow1/2 82.2%

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

      2. div-inv 82.1%

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

      3. unpow-prod-down 92.2%

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

      4. pow1/2 92.2%

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

   4. Applied egg-rr 92.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(\frac{1}{V \cdot \ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 92.2%

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

      2. associate-/r* 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-266}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-14}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -1e+185)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -1e-14)
     (* c0 (sqrt (/ A (* l V))))
     (if (<= (* l V) 2e-312)
       (* c0 (pow (* l (/ V A)) -0.5))
       (/ (* c0 (sqrt A)) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+185) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((l * V) <= -1e-14) {
		tmp = c0 * sqrt((A / (l * V)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * pow((l * (V / A)), -0.5);
	} else {
		tmp = (c0 * sqrt(A)) / sqrt((l * 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) :: tmp
    if ((l * v) <= (-1d+185)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((l * v) <= (-1d-14)) then
        tmp = c0 * sqrt((a / (l * v)))
    else if ((l * v) <= 2d-312) then
        tmp = c0 * ((l * (v / a)) ** (-0.5d0))
    else
        tmp = (c0 * sqrt(a)) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+185) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -1e-14) {
		tmp = c0 * Math.sqrt((A / (l * V)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * Math.pow((l * (V / A)), -0.5);
	} else {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -1e+185:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (l * V) <= -1e-14:
		tmp = c0 * math.sqrt((A / (l * V)))
	elif (l * V) <= 2e-312:
		tmp = c0 * math.pow((l * (V / A)), -0.5)
	else:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -1e+185)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -1e-14)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(l * V))));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 * (Float64(l * Float64(V / A)) ^ -0.5));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -1e+185)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -1e-14)
		tmp = c0 * sqrt((A / (l * V)));
	elseif ((l * V) <= 2e-312)
		tmp = c0 * ((l * (V / A)) ^ -0.5);
	else
		tmp = (c0 * sqrt(A)) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -1e+185], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-14], N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.9999999999999998e184

   1. Initial program 43.5%

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

      1. associate-/r* 58.8%

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

      2. sqrt-div 35.7%

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

      3. associate-*r/ 35.6%

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

   4. Applied egg-rr 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-/l* 35.6%

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

   6. Simplified 35.6%

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

   if -9.9999999999999998e184 < (*.f64 V l) < -9.99999999999999999e-15

   1. Initial program 92.8%

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

   if -9.99999999999999999e-15 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. times-frac 78.4%

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

   4. Applied egg-rr 78.4%

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

      1. clear-num 78.5%

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

      2. un-div-inv 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. associate-/r* 78.4%

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

      2. associate-*r/ 66.1%

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

      3. inv-pow 66.1%

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

      4. sqrt-pow1 67.0%

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

      5. metadata-eval 67.0%

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

      6. associate-*r/ 79.3%

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

   8. Applied egg-rr 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*l/ 67.0%

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

      3. associate-*r/ 78.6%

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

   10. Simplified 78.6%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. sqrt-div 92.1%

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

      3. associate-*l/ 90.5%

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

   4. Applied egg-rr 90.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-14}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-14}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -1e+185)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -1e-14)
     (* c0 (sqrt (/ A (* l V))))
     (if (<= (* l V) 2e-312)
       (* c0 (pow (* l (/ V A)) -0.5))
       (* (sqrt A) (/ c0 (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+185) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((l * V) <= -1e-14) {
		tmp = c0 * sqrt((A / (l * V)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * pow((l * (V / A)), -0.5);
	} else {
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
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) <= (-1d+185)) then
        tmp = (c0 / sqrt(l)) * sqrt((a / v))
    else if ((l * v) <= (-1d-14)) then
        tmp = c0 * sqrt((a / (l * v)))
    else if ((l * v) <= 2d-312) then
        tmp = c0 * ((l * (v / a)) ** (-0.5d0))
    else
        tmp = sqrt(a) * (c0 / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+185) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -1e-14) {
		tmp = c0 * Math.sqrt((A / (l * V)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * Math.pow((l * (V / A)), -0.5);
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -1e+185:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (l * V) <= -1e-14:
		tmp = c0 * math.sqrt((A / (l * V)))
	elif (l * V) <= 2e-312:
		tmp = c0 * math.pow((l * (V / A)), -0.5)
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -1e+185)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -1e-14)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(l * V))));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 * (Float64(l * Float64(V / A)) ^ -0.5));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -1e+185)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -1e-14)
		tmp = c0 * sqrt((A / (l * V)));
	elseif ((l * V) <= 2e-312)
		tmp = c0 * ((l * (V / A)) ^ -0.5);
	else
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -1e+185], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-14], N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.9999999999999998e184

   1. Initial program 43.5%

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

      1. associate-/r* 58.8%

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

      2. sqrt-div 35.7%

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

      3. associate-*r/ 35.6%

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

   4. Applied egg-rr 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-/l* 35.6%

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

   6. Simplified 35.6%

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

   if -9.9999999999999998e184 < (*.f64 V l) < -9.99999999999999999e-15

   1. Initial program 92.8%

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

   if -9.99999999999999999e-15 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. times-frac 78.4%

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

   4. Applied egg-rr 78.4%

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

      1. clear-num 78.5%

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

      2. un-div-inv 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. associate-/r* 78.4%

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

      2. associate-*r/ 66.1%

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

      3. inv-pow 66.1%

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

      4. sqrt-pow1 67.0%

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

      5. metadata-eval 67.0%

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

      6. associate-*r/ 79.3%

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

   8. Applied egg-rr 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*l/ 67.0%

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

      3. associate-*r/ 78.6%

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

   10. Simplified 78.6%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. sqrt-div 92.1%

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

      2. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-14}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+79}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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) -2e+79)
   (* c0 (* (sqrt (/ A V)) (sqrt (/ 1.0 l))))
   (if (<= (* l V) 2e-312)
     (/ c0 (sqrt (* V (/ l A))))
     (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e+79) {
		tmp = c0 * (sqrt((A / V)) * sqrt((1.0 / l)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((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) :: tmp
    if ((l * v) <= (-2d+79)) then
        tmp = c0 * (sqrt((a / v)) * sqrt((1.0d0 / l)))
    else if ((l * v) <= 2d-312) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((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 tmp;
	if ((l * V) <= -2e+79) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.sqrt((1.0 / l)));
	} else if ((l * V) <= 2e-312) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -2e+79:
		tmp = c0 * (math.sqrt((A / V)) * math.sqrt((1.0 / l)))
	elif (l * V) <= 2e-312:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -2e+79)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * sqrt(Float64(1.0 / l))));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(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)
	tmp = 0.0;
	if ((l * V) <= -2e+79)
		tmp = c0 * (sqrt((A / V)) * sqrt((1.0 / l)));
	elseif ((l * V) <= 2e-312)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((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_] := If[LessEqual[N[(l * V), $MachinePrecision], -2e+79], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.99999999999999993e79

   1. Initial program 56.6%

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

      1. pow1/2 56.6%

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

      2. associate-/r* 67.2%

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

      3. div-inv 67.2%

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

      4. unpow-prod-down 39.2%

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

      5. pow1/2 39.2%

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

   4. Applied egg-rr 39.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 39.2%

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

   6. Simplified 39.2%

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

   if -1.99999999999999993e79 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. times-frac 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-*r/ 79.7%

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

      4. frac-2neg 79.7%

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

      5. associate-*l/ 79.8%

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

      6. *-un-lft-identity 79.8%

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

      7. distribute-frac-neg2 79.8%

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

      8. sqrt-undiv 37.6%

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

      9. clear-num 37.6%

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

      10. un-div-inv 37.6%

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

      11. sqrt-undiv 79.8%

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

      12. add-sqr-sqrt 32.4%

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

      13. sqrt-unprod 22.8%

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

      14. sqr-neg 22.8%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.1%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 21.3%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r/ 77.3%

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

      2. *-commutative 77.3%

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

   8. Simplified 77.3%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. pow1/2 82.2%

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

      2. div-inv 82.1%

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

      3. unpow-prod-down 92.2%

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

      4. pow1/2 92.2%

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

   4. Applied egg-rr 92.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(\frac{1}{V \cdot \ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 92.2%

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

      2. associate-/r* 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+79}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

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+108)
   (/ (* c0 (sqrt (/ A V))) (sqrt l))
   (if (<= (* l V) 2e-312)
     (* c0 (sqrt (* (/ 1.0 V) (/ A l))))
     (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+108) {
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((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) :: tmp
    if ((l * v) <= (-5d+108)) then
        tmp = (c0 * sqrt((a / v))) / sqrt(l)
    else if ((l * v) <= 2d-312) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((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 tmp;
	if ((l * V) <= -5e+108) {
		tmp = (c0 * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e+108:
		tmp = (c0 * math.sqrt((A / V))) / math.sqrt(l)
	elif (l * V) <= 2e-312:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e+108)
		tmp = Float64(Float64(c0 * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(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)
	tmp = 0.0;
	if ((l * V) <= -5e+108)
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	elseif ((l * V) <= 2e-312)
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((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_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e+108], N[(N[(c0 * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999991e108

   1. Initial program 55.2%

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

      1. *-commutative 55.2%

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

      2. associate-/r* 66.7%

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

      3. sqrt-div 40.4%

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

      4. associate-*l/ 40.4%

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

   4. Applied egg-rr 40.4%

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

   if -4.99999999999999991e108 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. times-frac 76.4%

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

   4. Applied egg-rr 76.4%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. pow1/2 82.2%

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

      2. div-inv 82.1%

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

      3. unpow-prod-down 92.2%

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

      4. pow1/2 92.2%

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

   4. Applied egg-rr 92.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(\frac{1}{V \cdot \ell}\right)}^{0.5}\right)} \]
   5. Step-by-step derivation

      1. unpow1/2 92.2%

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

      2. associate-/r* 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e+108)
   (/ (* c0 (sqrt (/ A V))) (sqrt l))
   (if (<= (* l V) 2e-312)
     (* c0 (sqrt (* (/ 1.0 V) (/ A l))))
     (/ (* c0 (sqrt A)) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+108) {
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else {
		tmp = (c0 * sqrt(A)) / sqrt((l * 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) :: tmp
    if ((l * v) <= (-5d+108)) then
        tmp = (c0 * sqrt((a / v))) / sqrt(l)
    else if ((l * v) <= 2d-312) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else
        tmp = (c0 * sqrt(a)) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e+108) {
		tmp = (c0 * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((l * V) <= 2e-312) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e+108:
		tmp = (c0 * math.sqrt((A / V))) / math.sqrt(l)
	elif (l * V) <= 2e-312:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	else:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e+108)
		tmp = Float64(Float64(c0 * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(l * V) <= 2e-312)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e+108)
		tmp = (c0 * sqrt((A / V))) / sqrt(l);
	elseif ((l * V) <= 2e-312)
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	else
		tmp = (c0 * sqrt(A)) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e+108], N[(N[(c0 * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-312], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999991e108

   1. Initial program 55.2%

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

      1. *-commutative 55.2%

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

      2. associate-/r* 66.7%

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

      3. sqrt-div 40.4%

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

      4. associate-*l/ 40.4%

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

   4. Applied egg-rr 40.4%

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

   if -4.99999999999999991e108 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. times-frac 76.4%

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

   4. Applied egg-rr 76.4%

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

   if 2.0000000000019e-312 < (*.f64 V l)

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. sqrt-div 92.1%

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

      3. associate-*l/ 90.5%

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

   4. Applied egg-rr 90.5%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.5% 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 73.5%

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

3. Final simplification 73.5%

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

Reproduce

herbie shell --seed 2024143 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))