Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 85.4%

Time: 11.9s

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: 73.2% 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: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-121}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5.5e-121)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= l -5e-310)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (/ c0 (* (sqrt l) (sqrt (/ V A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5.5e-121) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if (l <= -5e-310) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5.5d-121)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if (l <= (-5d-310)) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5.5e-121) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if (l <= -5e-310) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5.5e-121:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif l <= -5e-310:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5.5e-121)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (l <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5.5e-121)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif (l <= -5e-310)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5.5e-121], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.50000000000000031e-121

   1. Initial program 77.3%

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

      1. clear-num 77.3%

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

      2. associate-/r/ 77.3%

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

      3. associate-/r* 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*r/ 74.4%

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

      3. div-inv 74.5%

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

      4. associate-/l/ 77.3%

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

      5. associate-/r* 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. frac-2neg 75.8%

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

      2. sqrt-div 38.0%

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

      3. distribute-neg-frac2 38.0%

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

   8. Applied egg-rr 38.0%

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

      1. distribute-frac-neg2 38.0%

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

      2. distribute-neg-frac 38.0%

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

   10. Simplified 38.0%

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

   if -5.50000000000000031e-121 < l < -4.999999999999985e-310

   1. Initial program 90.0%

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

      1. sqrt-div 59.8%

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

      2. div-inv 59.8%

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

   4. Applied egg-rr 59.8%

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

      1. associate-*r/ 59.8%

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

      2. *-rgt-identity 59.8%

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

   6. Simplified 59.8%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.9%

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

      1. clear-num 69.4%

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

      2. associate-/r/ 69.9%

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

      3. associate-/r* 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. sqrt-prod 39.7%

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

      2. associate-/r* 39.4%

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

      3. sqrt-div 39.4%

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

      4. metadata-eval 39.4%

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

      5. associate-/r/ 39.4%

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

      6. un-div-inv 39.4%

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

      7. sqrt-undiv 70.2%

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

      8. associate-*r/ 70.0%

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

      9. clear-num 69.8%

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

      10. clear-num 69.8%

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

      11. un-div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-/r/ 70.5%

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

      2. associate-*l/ 70.6%

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

      3. *-lft-identity 70.6%

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

      4. associate-/r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-/l* 70.2%

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

      7. associate-*l/ 70.0%

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

      8. associate-/r/ 75.5%

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

   8. Simplified 75.5%

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

      1. pow1/2 75.5%

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

      2. div-inv 75.5%

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

      3. unpow-prod-down 89.9%

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

      4. pow1/2 89.9%

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

      5. clear-num 91.0%

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

   10. Applied egg-rr 91.0%

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

       1. unpow1/2 91.0%

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

   12. Simplified 91.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-121}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 2e-258)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+240)
       (* c0 (sqrt (* A (/ (/ 1.0 l) V))))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 2e-258) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+240) {
		tmp = c0 * sqrt((A * ((1.0 / l) / V)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 2d-258) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+240) then
        tmp = c0 * sqrt((a * ((1.0d0 / l) / v)))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 2e-258) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+240) {
		tmp = c0 * Math.sqrt((A * ((1.0 / l) / V)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 2e-258:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+240:
		tmp = c0 * math.sqrt((A * ((1.0 / l) / V)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if (t_0 <= 2e-258)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+240)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / l) / V))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 2e-258)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+240)
		tmp = c0 * sqrt((A * ((1.0 / l) / V)));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-258], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+240], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / l), $MachinePrecision] / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999991e-258

   1. Initial program 70.4%

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

      1. clear-num 70.3%

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

      2. associate-/r/ 70.4%

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

      3. associate-/r* 70.7%

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

   4. Applied egg-rr 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r/ 75.7%

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

      3. div-inv 75.7%

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

      4. associate-/l/ 70.4%

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

      5. associate-/r* 73.5%

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

   6. Applied egg-rr 73.5%

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

   if 1.99999999999999991e-258 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000003e240

   1. Initial program 98.8%

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

      1. clear-num 97.9%

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

      2. associate-/r/ 98.8%

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

      3. associate-/r* 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r/ 81.3%

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

      3. div-inv 81.5%

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

      4. associate-/l/ 98.8%

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

      5. associate-/r* 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. div-inv 82.4%

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

      2. associate-/l* 99.0%

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

   8. Applied egg-rr 99.0%

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

   if 2.00000000000000003e240 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 53.3%

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

      1. clear-num 53.3%

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

      2. associate-/r/ 53.3%

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

      3. associate-/r* 53.3%

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

   4. Applied egg-rr 53.3%

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

      1. sqrt-prod 48.2%

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

      2. associate-/r* 48.3%

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

      3. sqrt-div 48.3%

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

      4. metadata-eval 48.3%

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

      5. associate-/r/ 48.4%

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

      6. un-div-inv 48.4%

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

      7. sqrt-undiv 55.9%

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

      8. associate-*r/ 61.4%

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

      9. clear-num 61.3%

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

      10. clear-num 61.3%

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

      11. un-div-inv 61.3%

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

   6. Applied egg-rr 61.3%

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

      1. associate-/r/ 61.3%

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

      2. associate-*l/ 61.3%

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

      3. *-lft-identity 61.3%

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

      4. associate-/r/ 62.5%

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

      5. *-commutative 62.5%

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

      6. associate-/l* 55.9%

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

      7. associate-*l/ 61.4%

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

      8. associate-/r/ 62.5%

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

   8. Simplified 62.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := c0 \cdot \sqrt{t\_0}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{c0}{{t\_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))) (t_1 (* c0 (sqrt t_0))))
   (if (<= t_1 2e-253)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_1 2e+257) (/ c0 (pow t_0 -0.5)) (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * sqrt(t_0);
	double tmp;
	if (t_1 <= 2e-253) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_1 <= 2e+257) {
		tmp = c0 / pow(t_0, -0.5);
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (l * v)
    t_1 = c0 * sqrt(t_0)
    if (t_1 <= 2d-253) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_1 <= 2d+257) then
        tmp = c0 / (t_0 ** (-0.5d0))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * Math.sqrt(t_0);
	double tmp;
	if (t_1 <= 2e-253) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_1 <= 2e+257) {
		tmp = c0 / Math.pow(t_0, -0.5);
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = c0 * math.sqrt(t_0)
	tmp = 0
	if t_1 <= 2e-253:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_1 <= 2e+257:
		tmp = c0 / math.pow(t_0, -0.5)
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	t_1 = Float64(c0 * sqrt(t_0))
	tmp = 0.0
	if (t_1 <= 2e-253)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_1 <= 2e+257)
		tmp = Float64(c0 / (t_0 ^ -0.5));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	t_1 = c0 * sqrt(t_0);
	tmp = 0.0;
	if (t_1 <= 2e-253)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_1 <= 2e+257)
		tmp = c0 / (t_0 ^ -0.5);
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-253], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+257], N[(c0 / N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e-253

   1. Initial program 70.7%

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

      1. clear-num 70.7%

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

      2. associate-/r/ 70.7%

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

      3. associate-/r* 71.1%

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

   4. Applied egg-rr 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*r/ 76.0%

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

      3. div-inv 76.0%

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

      4. associate-/l/ 70.7%

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

      5. associate-/r* 73.8%

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

   6. Applied egg-rr 73.8%

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

   if 2.0000000000000001e-253 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000006e257

   1. Initial program 98.8%

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

      1. clear-num 97.9%

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

      2. associate-/r/ 98.8%

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

      3. associate-/r* 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. sqrt-prod 43.1%

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

      2. associate-/r* 43.1%

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

      3. sqrt-div 43.1%

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

      4. metadata-eval 43.1%

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

      5. associate-/r/ 43.1%

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

      6. un-div-inv 43.0%

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

      7. sqrt-undiv 98.0%

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

      8. associate-*r/ 82.8%

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

      9. clear-num 82.8%

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

      10. clear-num 82.0%

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

      11. un-div-inv 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. associate-/r/ 82.1%

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

      2. associate-*l/ 82.1%

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

      3. *-lft-identity 82.1%

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

      4. associate-/r/ 81.6%

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

      5. *-commutative 81.6%

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

      6. associate-/l* 98.0%

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

      7. associate-*l/ 82.8%

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

      8. associate-/r/ 80.3%

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

   8. Simplified 80.3%

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

      1. clear-num 80.3%

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

      2. inv-pow 80.3%

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

      3. metadata-eval 80.3%

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

      4. pow-prod-up 80.3%

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

      5. sqrt-unprod 80.9%

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

      6. add-sqr-sqrt 81.1%

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

      7. associate-/l/ 98.8%

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

   10. Applied egg-rr 98.8%

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

   if 2.00000000000000006e257 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 51.7%

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

      1. clear-num 51.7%

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

      2. associate-/r/ 51.7%

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

      3. associate-/r* 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. sqrt-prod 46.6%

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

      2. associate-/r* 46.6%

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

      3. sqrt-div 46.6%

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

      4. metadata-eval 46.6%

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

      5. associate-/r/ 46.7%

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

      6. un-div-inv 46.7%

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

      7. sqrt-undiv 54.4%

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

      8. associate-*r/ 60.1%

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

      9. clear-num 60.0%

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

      10. clear-num 60.0%

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

      11. un-div-inv 60.0%

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

   6. Applied egg-rr 60.0%

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

      1. associate-/r/ 60.0%

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

      2. associate-*l/ 60.0%

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

      3. *-lft-identity 60.0%

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

      4. associate-/r/ 61.2%

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

      5. *-commutative 61.2%

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

      6. associate-/l* 54.4%

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

      7. associate-*l/ 60.1%

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

      8. associate-/r/ 61.2%

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

   8. Simplified 61.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{\ell \cdot V}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+210}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 4e+210)))
     (* 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 <= 4e+210)) {
		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 <= 4d+210))) 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 <= 4e+210)) {
		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 <= 4e+210):
		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 <= 4e+210))
		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 <= 4e+210)))
		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, 4e+210]], $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 3.99999999999999971e210 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.6%

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

      1. associate-/r* 72.3%

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

   3. Simplified 72.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999971e210

   1. Initial program 98.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 4 \cdot 10^{+210}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+240) t_0 (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+240) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+240) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+240) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+240:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(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 <= 2e+240)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+240)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(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, 2e+240], t$95$0, N[(c0 / N[Sqrt[N[(l / N[(A / V), $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 68.6%

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

      1. associate-/r* 74.3%

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

   3. Simplified 74.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000003e240

   1. Initial program 99.0%

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

   if 2.00000000000000003e240 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 53.3%

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

      1. clear-num 53.3%

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

      2. associate-/r/ 53.3%

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

      3. associate-/r* 53.3%

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

   4. Applied egg-rr 53.3%

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

      1. sqrt-prod 48.2%

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

      2. associate-/r* 48.3%

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

      3. sqrt-div 48.3%

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

      4. metadata-eval 48.3%

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

      5. associate-/r/ 48.4%

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

      6. un-div-inv 48.4%

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

      7. sqrt-undiv 55.9%

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

      8. associate-*r/ 61.4%

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

      9. clear-num 61.3%

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

      10. clear-num 61.3%

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

      11. un-div-inv 61.3%

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

   6. Applied egg-rr 61.3%

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

      1. associate-/r/ 61.3%

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

      2. associate-*l/ 61.3%

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

      3. *-lft-identity 61.3%

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

      4. associate-/r/ 62.5%

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

      5. *-commutative 62.5%

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

      6. associate-/l* 55.9%

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

      7. associate-*l/ 61.4%

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

      8. associate-/r/ 62.5%

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

   8. Simplified 62.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+111}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-V}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e+111)
   (* c0 (/ (sqrt (/ A (- V))) (sqrt (- l))))
   (if (<= l -5e-310)
     (* c0 (/ (sqrt A) (sqrt (* l V))))
     (/ c0 (* (sqrt l) (sqrt (/ V A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e+111) {
		tmp = c0 * (sqrt((A / -V)) / sqrt(-l));
	} else if (l <= -5e-310) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d+111)) then
        tmp = c0 * (sqrt((a / -v)) / sqrt(-l))
    else if (l <= (-5d-310)) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e+111) {
		tmp = c0 * (Math.sqrt((A / -V)) / Math.sqrt(-l));
	} else if (l <= -5e-310) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e+111:
		tmp = c0 * (math.sqrt((A / -V)) / math.sqrt(-l))
	elif l <= -5e-310:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e+111)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-V))) / sqrt(Float64(-l))));
	elseif (l <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e+111)
		tmp = c0 * (sqrt((A / -V)) / sqrt(-l));
	elseif (l <= -5e-310)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e+111], N[(c0 * N[(N[Sqrt[N[(A / (-V)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.9999999999999997e111

   1. Initial program 70.7%

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

      1. associate-/r* 69.2%

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

      2. frac-2neg 69.2%

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

      3. sqrt-div 80.1%

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

      4. distribute-neg-frac2 80.1%

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

   4. Applied egg-rr 80.1%

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

      1. distribute-frac-neg2 80.1%

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

      2. distribute-frac-neg 80.1%

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

   6. Simplified 80.1%

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

   if -4.9999999999999997e111 < l < -4.999999999999985e-310

   1. Initial program 88.5%

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

      1. sqrt-div 49.0%

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

      2. div-inv 49.0%

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

   4. Applied egg-rr 49.0%

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

      1. associate-*r/ 49.0%

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

      2. *-rgt-identity 49.0%

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

   6. Simplified 49.0%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.9%

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

      1. clear-num 69.4%

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

      2. associate-/r/ 69.9%

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

      3. associate-/r* 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. sqrt-prod 39.7%

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

      2. associate-/r* 39.4%

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

      3. sqrt-div 39.4%

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

      4. metadata-eval 39.4%

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

      5. associate-/r/ 39.4%

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

      6. un-div-inv 39.4%

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

      7. sqrt-undiv 70.2%

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

      8. associate-*r/ 70.0%

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

      9. clear-num 69.8%

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

      10. clear-num 69.8%

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

      11. un-div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-/r/ 70.5%

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

      2. associate-*l/ 70.6%

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

      3. *-lft-identity 70.6%

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

      4. associate-/r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-/l* 70.2%

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

      7. associate-*l/ 70.0%

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

      8. associate-/r/ 75.5%

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

   8. Simplified 75.5%

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

      1. pow1/2 75.5%

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

      2. div-inv 75.5%

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

      3. unpow-prod-down 89.9%

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

      4. pow1/2 89.9%

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

      5. clear-num 91.0%

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

   10. Applied egg-rr 91.0%

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

       1. unpow1/2 91.0%

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

   12. Simplified 91.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+111}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-V}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 4.9 \cdot 10^{-242}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

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 (<= A 4.9e-242)
   (* c0 (sqrt (* (/ A V) (/ 1.0 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 (A <= 4.9e-242) {
		tmp = c0 * sqrt(((A / V) * (1.0 / 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 (a <= 4.9d-242) then
        tmp = c0 * sqrt(((a / v) * (1.0d0 / 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 (A <= 4.9e-242) {
		tmp = c0 * Math.sqrt(((A / V) * (1.0 / 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 A <= 4.9e-242:
		tmp = c0 * math.sqrt(((A / V) * (1.0 / 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 (A <= 4.9e-242)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) * Float64(1.0 / l))));
	else
		tmp = Float64(c0 * Float64(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 (A <= 4.9e-242)
		tmp = c0 * sqrt(((A / V) * (1.0 / 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[A, 4.9e-242], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 4.90000000000000002e-242

   1. Initial program 73.7%

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

      1. associate-/r* 76.5%

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

      2. div-inv 76.5%

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

   4. Applied egg-rr 76.5%

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

   if 4.90000000000000002e-242 < A

   1. Initial program 76.7%

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

      1. sqrt-div 85.3%

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

      2. div-inv 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. associate-*r/ 85.3%

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

      2. *-rgt-identity 85.3%

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

   6. Simplified 85.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 4.9 \cdot 10^{-242}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 80.3%

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

      1. sqrt-div 44.6%

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

      2. div-inv 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. associate-*r/ 44.6%

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

      2. *-rgt-identity 44.6%

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

   6. Simplified 44.6%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.9%

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

      1. associate-/r* 75.2%

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

      2. sqrt-div 89.9%

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

      3. div-inv 89.8%

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

   4. Applied egg-rr 89.8%

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

      1. associate-*r/ 89.9%

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

      2. *-rgt-identity 89.9%

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

   6. Simplified 89.9%

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

4. Final simplification 67.2%

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

Alternative 9: 84.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 80.3%

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

      1. sqrt-div 44.6%

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

      2. div-inv 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. associate-*r/ 44.6%

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

      2. *-rgt-identity 44.6%

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

   6. Simplified 44.6%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.9%

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

      1. clear-num 69.4%

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

      2. associate-/r/ 69.9%

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

      3. associate-/r* 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. sqrt-prod 39.7%

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

      2. associate-/r* 39.4%

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

      3. sqrt-div 39.4%

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

      4. metadata-eval 39.4%

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

      5. associate-/r/ 39.4%

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

      6. un-div-inv 39.4%

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

      7. sqrt-undiv 70.2%

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

      8. associate-*r/ 70.0%

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

      9. clear-num 69.8%

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

      10. clear-num 69.8%

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

      11. un-div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-/r/ 70.5%

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

      2. associate-*l/ 70.6%

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

      3. *-lft-identity 70.6%

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

      4. associate-/r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-/l* 70.2%

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

      7. associate-*l/ 70.0%

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

      8. associate-/r/ 75.5%

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

   8. Simplified 75.5%

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

      1. pow1/2 75.5%

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

      2. div-inv 75.5%

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

      3. unpow-prod-down 89.9%

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

      4. pow1/2 89.9%

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

      5. clear-num 91.0%

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

   10. Applied egg-rr 91.0%

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

       1. unpow1/2 91.0%

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

   12. Simplified 91.0%

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

4. Final simplification 67.8%

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

Alternative 10: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

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 (/ A (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+251) (* c0 (sqrt 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 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+251) {
		tmp = c0 * sqrt(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 = a / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+251) then
        tmp = c0 * sqrt(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 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+251) {
		tmp = c0 * Math.sqrt(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 = A / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+251:
		tmp = c0 * math.sqrt(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(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+251)
		tmp = Float64(c0 * sqrt(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 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+251)
		tmp = c0 * sqrt(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[(A / N[(l * V), $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, 5e+251], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 37.6%

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

      1. associate-/r* 57.8%

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

   3. Simplified 57.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.0000000000000005e251

   1. Initial program 99.3%

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

   if 5.0000000000000005e251 < (/.f64 A (*.f64 V l))

   1. Initial program 45.9%

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

      1. clear-num 45.9%

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

      2. associate-/r/ 45.9%

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

      3. associate-/r* 45.9%

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

   4. Applied egg-rr 45.9%

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

      1. sqrt-prod 35.1%

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

      2. associate-/r* 35.1%

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

      3. sqrt-div 35.0%

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

      4. metadata-eval 35.0%

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

      5. associate-/r/ 35.1%

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

      6. un-div-inv 35.1%

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

      7. sqrt-undiv 47.4%

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

      8. associate-*r/ 52.1%

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

      9. clear-num 52.1%

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

      10. un-div-inv 52.1%

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

   6. Applied egg-rr 52.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-318}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 2e-318)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+292)
       (/ c0 (sqrt (/ (* l V) A)))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+292) {
		tmp = c0 / sqrt(((l * V) / A));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (l * v)
    if (t_0 <= 2d-318) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+292) then
        tmp = c0 / sqrt(((l * v) / a))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+292) {
		tmp = c0 / Math.sqrt(((l * V) / A));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 2e-318:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+292:
		tmp = c0 / math.sqrt(((l * V) / A))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 2e-318)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+292)
		tmp = Float64(c0 / sqrt(Float64(Float64(l * V) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 2e-318)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+292)
		tmp = c0 / sqrt(((l * V) / A));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-318], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+292], N[(c0 / N[Sqrt[N[(N[(l * V), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000024e-318

   1. Initial program 38.0%

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

      1. associate-/r* 57.7%

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

   3. Simplified 57.7%

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

   if 2.0000024e-318 < (/.f64 A (*.f64 V l)) < 2e292

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.6%

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

      3. associate-/r* 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. sqrt-prod 50.4%

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

      2. associate-/r* 50.3%

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

      3. sqrt-div 50.3%

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

      4. metadata-eval 50.3%

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

      5. associate-/r/ 50.3%

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

      6. un-div-inv 50.3%

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

      7. sqrt-undiv 99.7%

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

      8. associate-*r/ 86.4%

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

      9. clear-num 86.2%

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

      10. clear-num 85.9%

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

      11. un-div-inv 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. associate-/r/ 87.3%

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

      2. associate-*l/ 87.4%

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

      3. *-lft-identity 87.4%

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

      4. associate-/r/ 90.5%

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

      5. *-commutative 90.5%

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

      6. associate-/l* 99.7%

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

      7. associate-*l/ 86.4%

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

      8. associate-/r/ 89.1%

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

   8. Simplified 89.1%

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

   9. Taylor expanded in l around 0 99.7%

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

   if 2e292 < (/.f64 A (*.f64 V l))

   1. Initial program 38.1%

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

      1. clear-num 38.1%

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

      2. associate-/r/ 38.1%

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

      3. associate-/r* 38.1%

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

   4. Applied egg-rr 38.1%

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

      1. sqrt-prod 34.0%

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

      2. associate-/r* 34.0%

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

      3. sqrt-div 34.0%

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

      4. metadata-eval 34.0%

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

      5. associate-/r/ 34.0%

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

      6. un-div-inv 34.0%

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

      7. sqrt-undiv 39.8%

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

      8. associate-*r/ 51.1%

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

      9. clear-num 51.2%

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

      10. clear-num 51.2%

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

      11. un-div-inv 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. associate-/r/ 51.1%

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

      2. associate-*l/ 51.1%

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

      3. *-lft-identity 51.1%

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

      4. associate-/r/ 51.8%

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

      5. *-commutative 51.8%

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

      6. associate-/l* 39.8%

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

      7. associate-*l/ 51.1%

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

      8. associate-/r/ 51.8%

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

   8. Simplified 51.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 1.0× speedup

Math

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

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 75.1%

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

3. Final simplification 75.1%

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

Reproduce

herbie shell --seed 2024073 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))