Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 90.6%

Time: 11.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 90.6% accurate, 0.3× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (* c0 (/ (/ 1.0 (/ (sqrt (- V)) (sqrt (- A)))) (sqrt l)))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
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 <= (-2d-310)) then
        tmp = c0 * ((1.0d0 / (sqrt(-v) / sqrt(-a))) / sqrt(l))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

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

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = c0 * ((1.0 / (math.sqrt(-V) / math.sqrt(-A))) / math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(c0 * Float64(Float64(1.0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(-A)))) / sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = c0 * ((1.0 / (sqrt(-V) / sqrt(-A))) / sqrt(l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(c0 * N[(N[(1.0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 78.6%

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

      1. associate-/r* 78.4%

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

      2. sqrt-div 49.2%

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

      3. clear-num 49.1%

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

      4. sqrt-div 49.3%

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

      5. metadata-eval 49.3%

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

      6. associate-/l/ 49.2%

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

   3. Applied egg-rr 49.2%

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

      1. associate-/l/ 49.3%

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

   5. Simplified 49.3%

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

      1. frac-2neg 49.3%

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

      2. sqrt-div 53.5%

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

   7. Applied egg-rr 53.5%

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

   if -1.999999999999994e-310 < A

   1. Initial program 68.5%

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

      1. sqrt-div 86.6%

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

      2. associate-*r/ 83.6%

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

   3. Applied egg-rr 83.6%

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

      1. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

4. Final simplification 69.6%

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

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-290)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 4e+292) t_0 (* c0 (/ 1.0 (sqrt (* l (/ V A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-290) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 4e+292) {
		tmp = t_0;
	} else {
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-290) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 4d+292) then
        tmp = t_0
    else
        tmp = c0 * (1.0d0 / sqrt((l * (v / a))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-290) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 4e+292) {
		tmp = t_0;
	} else {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-290:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 4e+292:
		tmp = t_0
	else:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-290)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 4e+292)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(l * Float64(V / A)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-290)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 4e+292)
		tmp = t_0;
	else
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-290], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+292], t$95$0, N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e-290

   1. Initial program 68.5%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

      3. div-inv 69.1%

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

      4. associate-*l* 68.7%

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

   3. Applied egg-rr 68.7%

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

      1. frac-times 68.5%

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

      2. metadata-eval 68.5%

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

      3. div-inv 68.5%

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

      4. sqrt-div 41.6%

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

      5. clear-num 41.5%

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

      6. div-inv 41.6%

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

      7. sqrt-undiv 67.9%

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

      8. *-un-lft-identity 67.9%

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

      9. times-frac 70.3%

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

      10. /-rgt-identity 70.3%

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

   5. Applied egg-rr 70.3%

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

   if 2.0000000000000001e-290 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000001e292

   1. Initial program 99.4%

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

   if 4.0000000000000001e292 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 45.1%

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

      1. clear-num 45.1%

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

      2. sqrt-div 49.1%

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

      3. metadata-eval 49.1%

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

      4. *-commutative 49.1%

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

      5. associate-/l* 55.3%

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

      6. div-inv 55.2%

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

      7. clear-num 55.4%

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

   3. Applied egg-rr 55.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 3: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-290} \lor \neg \left(t_0 \leq 2 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 2e-290) (not (<= t_0 2e+265)))
     (/ c0 (sqrt (* V (/ l A))))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-290) || !(t_0 <= 2e+265)) {
		tmp = c0 / sqrt((V * (l / A)));
	} 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 / (v * l)))
    if ((t_0 <= 2d-290) .or. (.not. (t_0 <= 2d+265))) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-290) || !(t_0 <= 2e+265)) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 2e-290) or not (t_0 <= 2e+265):
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 2e-290) || !(t_0 <= 2e+265))
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 2e-290) || ~((t_0 <= 2e+265)))
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-290], N[Not[LessEqual[t$95$0, 2e+265]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e-290 or 2.00000000000000013e265 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.4%

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

      1. associate-/r* 67.0%

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

      2. div-inv 66.9%

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

      3. div-inv 66.9%

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

      4. associate-*l* 65.5%

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

   3. Applied egg-rr 65.5%

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

      1. frac-times 65.4%

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

      2. metadata-eval 65.4%

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

      3. div-inv 65.4%

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

      4. sqrt-div 39.7%

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

      5. clear-num 39.7%

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

      6. div-inv 39.7%

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

      7. sqrt-undiv 65.5%

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

      8. *-un-lft-identity 65.5%

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

      9. times-frac 67.4%

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

      10. /-rgt-identity 67.4%

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

   5. Applied egg-rr 67.4%

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

   if 2.0000000000000001e-290 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000013e265

   1. Initial program 99.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-290} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 4: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-290)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 2e+265) t_0 (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-290) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 2e+265) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
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 / (v * l)))
    if (t_0 <= 2d-290) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 2d+265) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-290) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 2e+265) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-290:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 2e+265:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-290)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 2e+265)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-290)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 2e+265)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-290], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+265], t$95$0, N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e-290

   1. Initial program 68.5%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

      3. div-inv 69.1%

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

      4. associate-*l* 68.7%

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

   3. Applied egg-rr 68.7%

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

      1. frac-times 68.5%

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

      2. metadata-eval 68.5%

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

      3. div-inv 68.5%

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

      4. sqrt-div 41.6%

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

      5. clear-num 41.5%

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

      6. div-inv 41.6%

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

      7. sqrt-undiv 67.9%

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

      8. *-un-lft-identity 67.9%

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

      9. times-frac 70.3%

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

      10. /-rgt-identity 70.3%

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

   5. Applied egg-rr 70.3%

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

   if 2.0000000000000001e-290 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000013e265

   1. Initial program 99.4%

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

   if 2.00000000000000013e265 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 51.3%

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

      1. associate-/r* 56.9%

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

      2. div-inv 56.8%

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

      3. div-inv 56.8%

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

      4. associate-*l* 51.3%

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

   3. Applied egg-rr 51.3%

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

      1. frac-times 51.3%

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

      2. metadata-eval 51.3%

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

      3. div-inv 51.3%

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

      4. sqrt-div 31.2%

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

      5. clear-num 31.2%

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

      6. div-inv 31.3%

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

      7. clear-num 31.3%

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

      8. sqrt-undiv 54.8%

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

      9. *-commutative 54.8%

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

      10. associate-*r/ 60.3%

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

      11. associate-*r/ 54.8%

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

      12. *-commutative 54.8%

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

      13. *-un-lft-identity 54.8%

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

      14. times-frac 54.1%

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

      15. /-rgt-identity 54.1%

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

   5. Applied egg-rr 54.1%

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

      1. associate-/r/ 54.3%

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

      2. associate-*l/ 54.3%

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

      3. *-lft-identity 54.3%

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

      4. associate-*r/ 54.9%

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

      5. associate-/l* 54.3%

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

   7. Simplified 54.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 5: 90.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
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 <= (-2d-310)) then
        tmp = c0 * (sqrt(-a) / (sqrt(-v) * sqrt(l)))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 78.6%

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

      1. associate-/r* 78.4%

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

      2. sqrt-div 49.2%

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

      3. frac-2neg 49.2%

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

      4. sqrt-div 53.5%

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

      5. associate-/l/ 53.5%

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

   3. Applied egg-rr 53.5%

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

   if -1.999999999999994e-310 < A

   1. Initial program 68.5%

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

      1. sqrt-div 86.6%

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

      2. associate-*r/ 83.6%

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

   3. Applied egg-rr 83.6%

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

      1. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

4. Final simplification 69.6%

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

Alternative 6: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-132}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -2e+215)
     t_0
     (if (<= (* V l) -4e-132)
       (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
       (if (<= (* V l) 0.0) t_0 (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+215) {
		tmp = t_0;
	} else if ((V * l) <= -4e-132) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-2d+215)) then
        tmp = t_0
    else if ((v * l) <= (-4d-132)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -2e+215) {
		tmp = t_0;
	} else if ((V * l) <= -4e-132) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -2e+215:
		tmp = t_0
	elif (V * l) <= -4e-132:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -2e+215)
		tmp = t_0;
	elseif (Float64(V * l) <= -4e-132)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -2e+215)
		tmp = t_0;
	elseif ((V * l) <= -4e-132)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+215], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -4e-132], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.99999999999999981e215 or -3.9999999999999999e-132 < (*.f64 V l) < 0.0

   1. Initial program 59.0%

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

      1. associate-/r* 75.5%

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

      2. div-inv 75.4%

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

      3. div-inv 75.4%

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

      4. associate-*l* 58.9%

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

   3. Applied egg-rr 58.9%

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

      1. frac-times 59.0%

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

      2. metadata-eval 59.0%

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

      3. div-inv 59.0%

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

      4. clear-num 59.0%

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

      5. *-commutative 59.0%

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

      6. associate-*r/ 75.5%

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

      7. sqrt-div 76.0%

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

      8. metadata-eval 76.0%

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

      9. sqrt-prod 50.4%

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

      10. associate-/l/ 50.5%

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

      11. add-sqr-sqrt 50.2%

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

      12. *-un-lft-identity 50.2%

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

      13. times-frac 50.3%

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

   5. Applied egg-rr 50.2%

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

      1. associate-*r/ 50.2%

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

      2. /-rgt-identity 50.2%

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

      3. rem-square-sqrt 50.4%

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

   7. Simplified 50.4%

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

   if -1.99999999999999981e215 < (*.f64 V l) < -3.9999999999999999e-132

   1. Initial program 95.1%

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

      1. associate-/r* 79.6%

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

      2. div-inv 79.6%

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

      3. div-inv 79.6%

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

      4. associate-*l* 95.2%

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

   3. Applied egg-rr 95.2%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 71.4%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

      1. associate-/r/ 89.6%

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

   7. Applied egg-rr 89.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+215}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-132}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-132}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -2e+215)
     t_0
     (if (<= (* V l) -4e-132)
       (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
       (if (<= (* V l) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+215) {
		tmp = t_0;
	} else if ((V * l) <= -4e-132) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-2d+215)) then
        tmp = t_0
    else if ((v * l) <= (-4d-132)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -2e+215) {
		tmp = t_0;
	} else if ((V * l) <= -4e-132) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -2e+215:
		tmp = t_0
	elif (V * l) <= -4e-132:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -2e+215)
		tmp = t_0;
	elseif (Float64(V * l) <= -4e-132)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -2e+215)
		tmp = t_0;
	elseif ((V * l) <= -4e-132)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+215], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -4e-132], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.99999999999999981e215 or -3.9999999999999999e-132 < (*.f64 V l) < 0.0

   1. Initial program 59.0%

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

      1. associate-/r* 75.5%

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

      2. div-inv 75.4%

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

      3. div-inv 75.4%

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

      4. associate-*l* 58.9%

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

   3. Applied egg-rr 58.9%

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

      1. frac-times 59.0%

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

      2. metadata-eval 59.0%

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

      3. div-inv 59.0%

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

      4. clear-num 59.0%

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

      5. *-commutative 59.0%

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

      6. associate-*r/ 75.5%

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

      7. sqrt-div 76.0%

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

      8. metadata-eval 76.0%

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

      9. sqrt-prod 50.4%

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

      10. associate-/l/ 50.5%

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

      11. add-sqr-sqrt 50.2%

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

      12. *-un-lft-identity 50.2%

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

      13. times-frac 50.3%

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

   5. Applied egg-rr 50.2%

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

      1. associate-*r/ 50.2%

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

      2. /-rgt-identity 50.2%

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

      3. rem-square-sqrt 50.4%

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

   7. Simplified 50.4%

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

   if -1.99999999999999981e215 < (*.f64 V l) < -3.9999999999999999e-132

   1. Initial program 95.1%

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

      1. associate-/r* 79.6%

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

      2. div-inv 79.6%

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

      3. div-inv 79.6%

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

      4. associate-*l* 95.2%

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

   3. Applied egg-rr 95.2%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 71.4%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-/l* 89.6%

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

      3. associate-/r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+215}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-132}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 8: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt l) (sqrt (/ A V))))))
   (if (<= (* V l) -2e+278)
     t_0
     (if (<= (* V l) -4e-128)
       (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
       (if (<= (* V l) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -2e+278) {
		tmp = t_0;
	} else if ((V * l) <= -4e-128) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) / sqrt((a / v)))
    if ((v * l) <= (-2d+278)) then
        tmp = t_0
    else if ((v * l) <= (-4d-128)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -2e+278) {
		tmp = t_0;
	} else if ((V * l) <= -4e-128) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -2e+278:
		tmp = t_0
	elif (V * l) <= -4e-128:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -2e+278)
		tmp = t_0;
	elseif (Float64(V * l) <= -4e-128)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -2e+278)
		tmp = t_0;
	elseif ((V * l) <= -4e-128)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+278], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -4e-128], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.99999999999999993e278 or -4.00000000000000022e-128 < (*.f64 V l) < 0.0

   1. Initial program 57.2%

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

      1. associate-/r* 74.0%

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

      2. sqrt-div 48.5%

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

      3. associate-*r/ 46.2%

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

   3. Applied egg-rr 46.2%

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

      1. associate-/l* 48.6%

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

   5. Simplified 48.6%

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

   if -1.99999999999999993e278 < (*.f64 V l) < -4.00000000000000022e-128

   1. Initial program 96.5%

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

      1. associate-/r* 81.3%

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

      2. div-inv 81.2%

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

      3. div-inv 81.2%

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

      4. associate-*l* 96.6%

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

   3. Applied egg-rr 96.6%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 71.4%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-/l* 89.6%

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

      3. associate-/r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 9: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt l) (sqrt (/ A V))))))
   (if (<= (* V l) -2e+278)
     t_0
     (if (<= (* V l) -4e-128)
       (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
       (if (<= (* V l) 0.0) t_0 (/ c0 (/ (sqrt (* V l)) (sqrt A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -2e+278) {
		tmp = t_0;
	} else if ((V * l) <= -4e-128) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
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(l) / sqrt((a / v)))
    if ((v * l) <= (-2d+278)) then
        tmp = t_0
    else if ((v * l) <= (-4d-128)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -2e+278) {
		tmp = t_0;
	} else if ((V * l) <= -4e-128) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -2e+278:
		tmp = t_0
	elif (V * l) <= -4e-128:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -2e+278)
		tmp = t_0;
	elseif (Float64(V * l) <= -4e-128)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -2e+278)
		tmp = t_0;
	elseif ((V * l) <= -4e-128)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+278], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -4e-128], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.99999999999999993e278 or -4.00000000000000022e-128 < (*.f64 V l) < 0.0

   1. Initial program 57.2%

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

      1. associate-/r* 74.0%

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

      2. sqrt-div 48.5%

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

      3. associate-*r/ 46.2%

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

   3. Applied egg-rr 46.2%

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

      1. associate-/l* 48.6%

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

   5. Simplified 48.6%

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

   if -1.99999999999999993e278 < (*.f64 V l) < -4.00000000000000022e-128

   1. Initial program 96.5%

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

      1. associate-/r* 81.3%

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

      2. div-inv 81.2%

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

      3. div-inv 81.2%

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

      4. associate-*l* 96.6%

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

   3. Applied egg-rr 96.6%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 71.4%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 10: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-131}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) (- INFINITY))
     (/ (* c0 t_0) (sqrt l))
     (if (<= (* V l) -1e-131)
       (* c0 (/ (sqrt (- A)) (sqrt (* (- V) l))))
       (if (<= (* V l) 0.0)
         (/ c0 (/ (sqrt l) t_0))
         (/ c0 (/ (sqrt (* V l)) (sqrt A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0 * t_0) / sqrt(l);
	} else if ((V * l) <= -1e-131) {
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / t_0);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 * t_0) / Math.sqrt(l);
	} else if ((V * l) <= -1e-131) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((-V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0 * t_0) / math.sqrt(l)
	elif (V * l) <= -1e-131:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((-V * l)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / t_0)
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0 * t_0) / sqrt(l));
	elseif (Float64(V * l) <= -1e-131)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0 * t_0) / sqrt(l);
	elseif ((V * l) <= -1e-131)
		tmp = c0 * (sqrt(-A) / sqrt((-V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) / t_0);
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0 * t$95$0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-131], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 28.9%

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

      1. *-commutative 28.9%

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

      2. associate-/r* 84.5%

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

      3. sqrt-div 37.0%

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

      4. associate-*l/ 37.3%

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

   3. Applied egg-rr 37.3%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999999e-132

   1. Initial program 93.2%

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

      1. frac-2neg 93.2%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -9.9999999999999999e-132 < (*.f64 V l) < 0.0

   1. Initial program 64.2%

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

      1. associate-/r* 71.7%

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

      2. sqrt-div 52.7%

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

      3. associate-*r/ 51.0%

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

   3. Applied egg-rr 51.0%

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

      1. associate-/l* 52.7%

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

   5. Simplified 52.7%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 71.4%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 87.9%

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

   3. Applied egg-rr 87.9%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-131}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 11: 74.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		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 / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 68.3%

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

      1. associate-/r* 69.0%

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

   3. Simplified 69.0%

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

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

   1. Initial program 82.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 12: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-279}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 1.18e-279)
   (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 1.18e-279) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 1.18d-279) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 1.18e-279) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 1.18e-279:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 1.18e-279)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 1.18e-279)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 1.18e-279], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.1800000000000001e-279

   1. Initial program 75.9%

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

      1. associate-/r* 70.9%

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

      2. div-inv 70.9%

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

      3. div-inv 70.9%

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

      4. associate-*l* 76.1%

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

   3. Applied egg-rr 76.1%

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

   if 1.1800000000000001e-279 < l

   1. Initial program 71.1%

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

      1. associate-/r* 74.0%

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

      2. div-inv 74.0%

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

      3. div-inv 73.9%

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

      4. associate-*l* 71.1%

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

   3. Applied egg-rr 71.1%

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

      1. frac-times 71.1%

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

      2. metadata-eval 71.1%

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

      3. div-inv 71.1%

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

      4. clear-num 70.6%

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

      5. *-commutative 70.6%

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

      6. associate-*r/ 73.5%

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

      7. sqrt-div 73.7%

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

      8. metadata-eval 73.7%

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

      9. sqrt-prod 90.5%

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

      10. associate-/l/ 90.7%

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

      11. add-sqr-sqrt 90.4%

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

      12. *-un-lft-identity 90.4%

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

      13. times-frac 90.5%

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

   5. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.1%

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

      2. /-rgt-identity 90.1%

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

      3. rem-square-sqrt 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-279}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 13: 73.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

2. Final simplification 73.7%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))