Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.6% → 90.7%

Time: 15.9s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = c0 / (sqrt(l) / (sqrt(-a) / sqrt(-v)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 75.7%

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

      1. associate-/r* 74.0%

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

      2. sqrt-div 43.2%

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

      3. associate-*r/ 41.1%

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

   4. Applied egg-rr 41.1%

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

      1. associate-/l* 43.1%

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

   6. Simplified 43.1%

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

      1. frac-2neg 43.1%

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

      2. sqrt-div 50.8%

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

   8. Applied egg-rr 50.8%

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

   if -4.999999999999985e-310 < A

   1. Initial program 72.8%

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

      1. div-inv 72.2%

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

      2. sqrt-prod 85.0%

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

      3. associate-/r* 85.5%

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

   4. Applied egg-rr 85.5%

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

4. Final simplification 67.6%

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

Alternative 2: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-1}{V}}}{\sqrt{\frac{-\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\frac{\sqrt{\ell}}{{V}^{-0.5}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -5e-310)
   (* c0 (/ (sqrt (/ -1.0 V)) (sqrt (/ (- l) A))))
   (/ (* c0 (sqrt A)) (/ (sqrt l) (pow V -0.5)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * (sqrt((-1.0 / V)) / sqrt((-l / A)));
	} else {
		tmp = (c0 * sqrt(A)) / (sqrt(l) / pow(V, -0.5));
	}
	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 (v <= (-5d-310)) then
        tmp = c0 * (sqrt(((-1.0d0) / v)) / sqrt((-l / a)))
    else
        tmp = (c0 * sqrt(a)) / (sqrt(l) / (v ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * (Math.sqrt((-1.0 / V)) / Math.sqrt((-l / A)));
	} else {
		tmp = (c0 * Math.sqrt(A)) / (Math.sqrt(l) / Math.pow(V, -0.5));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -5e-310:
		tmp = c0 * (math.sqrt((-1.0 / V)) / math.sqrt((-l / A)))
	else:
		tmp = (c0 * math.sqrt(A)) / (math.sqrt(l) / math.pow(V, -0.5))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(-1.0 / V)) / sqrt(Float64(Float64(-l) / A))));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / Float64(sqrt(l) / (V ^ -0.5)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -5e-310)
		tmp = c0 * (sqrt((-1.0 / V)) / sqrt((-l / A)));
	else
		tmp = (c0 * sqrt(A)) / (sqrt(l) / (V ^ -0.5));
	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[V, -5e-310], N[(c0 * N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[((-l) / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / N[Power[V, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 72.6%

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

      1. clear-num 71.9%

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

      2. associate-/r/ 72.5%

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

      3. associate-/r* 72.6%

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

   4. Applied egg-rr 72.6%

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

      1. associate-*l/ 74.1%

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

      2. associate-*r/ 70.9%

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

      3. associate-*l/ 70.9%

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

      4. *-un-lft-identity 70.9%

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

   6. Applied egg-rr 70.9%

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

      1. associate-/l/ 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. frac-times 70.9%

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

      4. clear-num 70.9%

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

      5. div-inv 70.9%

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

      6. frac-2neg 70.9%

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

      7. sqrt-div 82.8%

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

      8. distribute-neg-frac 82.8%

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

      9. metadata-eval 82.8%

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

      10. distribute-neg-frac 82.8%

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

   8. Applied egg-rr 82.8%

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

   if -4.999999999999985e-310 < V

   1. Initial program 76.0%

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

      1. clear-num 75.8%

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

      2. associate-/r/ 75.4%

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

      3. associate-/r* 76.0%

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

   4. Applied egg-rr 76.0%

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

      1. sqrt-prod 44.0%

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

      2. *-commutative 44.0%

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

      3. associate-*r* 41.9%

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

      4. *-commutative 41.9%

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

      5. sqrt-div 47.9%

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

      6. associate-*r/ 47.2%

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

      7. *-commutative 47.2%

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

      8. inv-pow 47.2%

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

      9. sqrt-pow1 47.2%

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

      10. metadata-eval 47.2%

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

   6. Applied egg-rr 47.2%

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

      1. associate-/l* 47.9%

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

   8. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{A}}{\frac{\sqrt{\ell}}{{V}^{-0.5}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-1}{V}}}{\sqrt{\frac{-\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\frac{\sqrt{\ell}}{{V}^{-0.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-297}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-302}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \frac{{c0}^{2}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* l V) -5e-297)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* l V) 1e-302)
       (sqrt (* (/ A l) (/ (pow c0 2.0) V)))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((l * V) <= -5e-297) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 1e-302) {
		tmp = sqrt(((A / l) * (pow(c0, 2.0) / V)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((l * V) <= -5e-297) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 1e-302) {
		tmp = Math.sqrt(((A / l) * (Math.pow(c0, 2.0) / V)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (l * V) <= -5e-297:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 1e-302:
		tmp = math.sqrt(((A / l) * (math.pow(c0, 2.0) / V)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(l * V) <= -5e-297)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 1e-302)
		tmp = sqrt(Float64(Float64(A / l) * Float64((c0 ^ 2.0) / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((l * V) <= -5e-297)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 1e-302)
		tmp = sqrt(((A / l) * ((c0 ^ 2.0) / V)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-297], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-302], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(N[Power[c0, 2.0], $MachinePrecision] / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 36.6%

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

      1. associate-/r* 62.9%

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

      2. sqrt-div 52.2%

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

      3. associate-*r/ 52.1%

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

   4. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-/l* 52.0%

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

      3. associate-/r/ 52.2%

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

   6. Simplified 52.2%

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

   if -inf.0 < (*.f64 V l) < -5e-297

   1. Initial program 86.3%

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

      1. frac-2neg 86.3%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if -5e-297 < (*.f64 V l) < 9.9999999999999996e-303

   1. Initial program 54.6%

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

      1. clear-num 54.6%

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

      2. associate-/r/ 52.2%

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

      3. associate-/r* 52.2%

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

   4. Applied egg-rr 52.2%

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

      1. sqrt-prod 34.6%

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

      2. *-commutative 34.6%

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

      3. add-sqr-sqrt 20.3%

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

      4. sqrt-unprod 20.5%

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

      5. *-commutative 20.5%

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

      6. *-commutative 20.5%

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

      7. swap-sqr 20.2%

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

      8. swap-sqr 20.2%

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

      9. add-sqr-sqrt 20.2%

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

      10. add-sqr-sqrt 36.9%

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

      11. associate-/r* 36.9%

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

      12. div-inv 39.3%

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

      13. pow2 39.3%

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

   6. Applied egg-rr 39.3%

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

      1. associate-*l/ 39.4%

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

      2. *-commutative 39.4%

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

      3. times-frac 40.3%

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

   8. Simplified 40.3%

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

   if 9.9999999999999996e-303 < (*.f64 V l)

   1. Initial program 75.6%

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

      1. div-inv 75.6%

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

      2. sqrt-prod 89.6%

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

      3. associate-/r* 90.3%

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

   4. Applied egg-rr 90.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-297}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-302}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \frac{{c0}^{2}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{V}}{\ell}\\ \mathbf{if}\;A \leq -4.2 \cdot 10^{-132}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot t_0}\\ \mathbf{elif}\;A \leq 8.2 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{A} \cdot \left(c0 \cdot {\left(\ell \cdot V\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{t_0}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 V) l)))
   (if (<= A -4.2e-132)
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (if (<= A 1.3e-305)
       (* c0 (sqrt (* A t_0)))
       (if (<= A 8.2e-169)
         (* (sqrt A) (* c0 (pow (* l V) -0.5)))
         (* c0 (* (sqrt A) (sqrt t_0))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (1.0 / V) / l;
	double tmp;
	if (A <= -4.2e-132) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if (A <= 1.3e-305) {
		tmp = c0 * sqrt((A * t_0));
	} else if (A <= 8.2e-169) {
		tmp = sqrt(A) * (c0 * pow((l * V), -0.5));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(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 = (1.0d0 / v) / l
    if (a <= (-4.2d-132)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if (a <= 1.3d-305) then
        tmp = c0 * sqrt((a * t_0))
    else if (a <= 8.2d-169) then
        tmp = sqrt(a) * (c0 * ((l * v) ** (-0.5d0)))
    else
        tmp = c0 * (sqrt(a) * sqrt(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 = (1.0 / V) / l;
	double tmp;
	if (A <= -4.2e-132) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if (A <= 1.3e-305) {
		tmp = c0 * Math.sqrt((A * t_0));
	} else if (A <= 8.2e-169) {
		tmp = Math.sqrt(A) * (c0 * Math.pow((l * V), -0.5));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(t_0));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (1.0 / V) / l
	tmp = 0
	if A <= -4.2e-132:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif A <= 1.3e-305:
		tmp = c0 * math.sqrt((A * t_0))
	elif A <= 8.2e-169:
		tmp = math.sqrt(A) * (c0 * math.pow((l * V), -0.5))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(t_0))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(1.0 / V) / l)
	tmp = 0.0
	if (A <= -4.2e-132)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (A <= 1.3e-305)
		tmp = Float64(c0 * sqrt(Float64(A * t_0)));
	elseif (A <= 8.2e-169)
		tmp = Float64(sqrt(A) * Float64(c0 * (Float64(l * V) ^ -0.5)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(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 = (1.0 / V) / l;
	tmp = 0.0;
	if (A <= -4.2e-132)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif (A <= 1.3e-305)
		tmp = c0 * sqrt((A * t_0));
	elseif (A <= 8.2e-169)
		tmp = sqrt(A) * (c0 * ((l * V) ^ -0.5));
	else
		tmp = c0 * (sqrt(A) * sqrt(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[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[A, -4.2e-132], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.3e-305], N[(c0 * N[Sqrt[N[(A * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 8.2e-169], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 * N[Power[N[(l * V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -4.2000000000000002e-132

   1. Initial program 73.5%

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

      1. associate-/r* 74.9%

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

      2. sqrt-div 45.5%

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

      3. associate-*r/ 43.6%

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

   4. Applied egg-rr 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/l* 44.5%

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

      3. associate-/r/ 45.5%

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

   6. Simplified 45.5%

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

   if -4.2000000000000002e-132 < A < 1.3000000000000001e-305

   1. Initial program 82.2%

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

      1. clear-num 82.2%

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

      2. associate-/r/ 82.2%

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

      3. associate-/r* 82.3%

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

   4. Applied egg-rr 82.3%

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

   if 1.3000000000000001e-305 < A < 8.1999999999999996e-169

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. sqrt-div 92.0%

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

      3. associate-*l/ 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. div-inv 85.4%

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

      2. *-commutative 85.4%

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

      3. metadata-eval 85.4%

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

      4. sqrt-div 82.8%

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

      5. associate-/r* 82.7%

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

      6. associate-*r* 89.2%

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

      7. *-commutative 89.2%

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

      8. associate-*r* 89.3%

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

      9. associate-/r* 89.5%

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

      10. sqrt-div 92.0%

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

      11. metadata-eval 92.0%

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

      12. pow1/2 92.0%

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

      13. pow-flip 92.1%

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

      14. metadata-eval 92.1%

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

   6. Applied egg-rr 92.1%

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

   if 8.1999999999999996e-169 < A

   1. Initial program 72.6%

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

      1. div-inv 72.6%

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

      2. sqrt-prod 83.5%

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

      3. associate-/r* 84.3%

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

   4. Applied egg-rr 84.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.2 \cdot 10^{-132}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;A \leq 8.2 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{A} \cdot \left(c0 \cdot {\left(\ell \cdot V\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+234}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 4 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{\ell \cdot V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -1e+234)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* l V) 4e-314)
     (* c0 (sqrt (/ 1.0 (/ (* l V) A))))
     (* (sqrt A) (/ c0 (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+234) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((l * V) <= 4e-314) {
		tmp = c0 * sqrt((1.0 / ((l * V) / A)));
	} else {
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-1d+234)) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((l * v) <= 4d-314) then
        tmp = c0 * sqrt((1.0d0 / ((l * v) / a)))
    else
        tmp = sqrt(a) * (c0 / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+234) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((l * V) <= 4e-314) {
		tmp = c0 * Math.sqrt((1.0 / ((l * V) / A)));
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -1e+234:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (l * V) <= 4e-314:
		tmp = c0 * math.sqrt((1.0 / ((l * V) / A)))
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -1e+234)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(l * V) <= 4e-314)
		tmp = Float64(c0 * sqrt(Float64(1.0 / Float64(Float64(l * V) / A))));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -1e+234)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((l * V) <= 4e-314)
		tmp = c0 * sqrt((1.0 / ((l * V) / A)));
	else
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -1e+234], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 4e-314], N[(c0 * N[Sqrt[N[(1.0 / N[(N[(l * V), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.00000000000000002e234

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-/l/ 64.4%

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

   3. Simplified 64.4%

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

   if -1.00000000000000002e234 < (*.f64 V l) < 3.9999999999e-314

   1. Initial program 80.2%

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

      1. clear-num 80.2%

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

      2. associate-/r/ 79.5%

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

      3. associate-/r* 79.6%

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

   4. Applied egg-rr 79.6%

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

      1. associate-*l/ 74.1%

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

      2. associate-*r/ 72.9%

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

      3. frac-times 80.2%

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

      4. associate-/l* 80.2%

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

   6. Applied egg-rr 80.2%

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

   if 3.9999999999e-314 < (*.f64 V l)

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. sqrt-div 89.8%

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

      3. associate-*l/ 84.9%

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

   4. Applied egg-rr 84.9%

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

   6. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.25} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25} \cdot c0\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 75.2%

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

      2. associate-/l/ 71.6%

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

      3. associate-/l/ 75.0%

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

      4. pow-prod-up 75.2%

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

      5. metadata-eval 75.2%

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

      6. pow1/2 75.2%

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

      7. *-commutative 75.2%

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

      8. associate-/l/ 75.4%

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

      9. associate-/r* 75.3%

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

      10. sqrt-div 42.2%

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

      11. clear-num 42.2%

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

      12. div-inv 42.3%

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

      13. associate-/r/ 42.0%

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

      14. sqrt-div 49.9%

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

      15. frac-times 47.6%

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

      16. sqrt-prod 84.9%

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

      17. *-commutative 84.9%

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

   8. Applied egg-rr 84.9%

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

      1. associate-/l* 89.8%

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

      2. associate-/r/ 88.1%

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

   10. Simplified 88.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+234}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 4 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{\ell \cdot V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t_0 \leq 10^{-190}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (<= t_0 1e-190) (* c0 (sqrt (* (/ 1.0 V) (/ A l)))) t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 1e-190) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 1d-190) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 1e-190) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 1e-190:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if (t_0 <= 1e-190)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 1e-190)
		tmp = c0 * sqrt(((1.0 / V) * (A / 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[LessEqual[t$95$0, 1e-190], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e-190

   1. Initial program 69.1%

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

      1. *-un-lft-identity 69.1%

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

      2. times-frac 70.4%

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

   4. Applied egg-rr 70.4%

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

   if 1e-190 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 84.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 10^{-190}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-252}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{V}}{\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 4e-252) (* c0 (sqrt (/ (/ 1.0 V) (/ l A)))) t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 4e-252) {
		tmp = c0 * sqrt(((1.0 / V) / (l / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 4d-252) then
        tmp = c0 * sqrt(((1.0d0 / v) / (l / a)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 4e-252) {
		tmp = c0 * Math.sqrt(((1.0 / V) / (l / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 4e-252:
		tmp = c0 * math.sqrt(((1.0 / V) / (l / A)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(l * V))))
	tmp = 0.0
	if (t_0 <= 4e-252)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) / Float64(l / A))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (l * V)));
	tmp = 0.0;
	if (t_0 <= 4e-252)
		tmp = c0 * sqrt(((1.0 / V) / (l / A)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-252], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999977e-252

   1. Initial program 68.4%

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

      1. clear-num 67.9%

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

      2. associate-/r/ 68.4%

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

      3. associate-/r* 68.8%

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

   4. Applied egg-rr 68.8%

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

      1. associate-*l/ 71.9%

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

      2. associate-/l* 69.3%

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

   6. Applied egg-rr 69.3%

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

   if 3.99999999999999977e-252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 85.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 4 \cdot 10^{-252}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{V}}{\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.85 \cdot 10^{-133}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \left(c0 \cdot {\left(\ell \cdot V\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1.85e-133)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= A 1.3e-305)
     (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
     (* (sqrt A) (* c0 (pow (* l V) -0.5))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1.85e-133) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if (A <= 1.3e-305) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = sqrt(A) * (c0 * pow((l * V), -0.5));
	}
	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 <= (-1.85d-133)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if (a <= 1.3d-305) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else
        tmp = sqrt(a) * (c0 * ((l * v) ** (-0.5d0)))
    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 <= -1.85e-133) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if (A <= 1.3e-305) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = Math.sqrt(A) * (c0 * Math.pow((l * V), -0.5));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -1.85e-133:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif A <= 1.3e-305:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	else:
		tmp = math.sqrt(A) * (c0 * math.pow((l * V), -0.5))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -1.85e-133)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (A <= 1.3e-305)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(sqrt(A) * Float64(c0 * (Float64(l * V) ^ -0.5)));
	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 <= -1.85e-133)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif (A <= 1.3e-305)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	else
		tmp = sqrt(A) * (c0 * ((l * V) ^ -0.5));
	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, -1.85e-133], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.3e-305], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 * N[Power[N[(l * V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if A < -1.85000000000000018e-133

   1. Initial program 73.5%

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

      1. associate-/r* 74.9%

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

      2. sqrt-div 45.5%

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

      3. associate-*r/ 43.6%

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

   4. Applied egg-rr 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/l* 44.5%

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

      3. associate-/r/ 45.5%

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

   6. Simplified 45.5%

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

   if -1.85000000000000018e-133 < A < 1.3000000000000001e-305

   1. Initial program 82.2%

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

      1. clear-num 82.2%

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

      2. associate-/r/ 82.2%

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

      3. associate-/r* 82.3%

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

   4. Applied egg-rr 82.3%

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

   if 1.3000000000000001e-305 < A

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. sqrt-div 85.4%

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

      3. associate-*l/ 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. div-inv 80.9%

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

      2. *-commutative 80.9%

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

      3. metadata-eval 80.9%

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

      4. sqrt-div 80.3%

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

      5. associate-/r* 81.0%

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

      6. associate-*r* 85.4%

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

      7. *-commutative 85.4%

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

      8. associate-*r* 84.1%

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

      9. associate-/r* 84.1%

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

      10. sqrt-div 84.6%

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

      11. metadata-eval 84.6%

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

      12. pow1/2 84.6%

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

      13. pow-flip 84.7%

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

      14. metadata-eval 84.7%

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

   6. Applied egg-rr 84.7%

      \[\leadsto \color{blue}{\left(c0 \cdot {\left(V \cdot \ell\right)}^{-0.5}\right) \cdot \sqrt{A}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.85 \cdot 10^{-133}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \left(c0 \cdot {\left(\ell \cdot V\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-/l/ 71.5%

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

   3. Simplified 71.5%

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

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

   1. Initial program 86.0%

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

4. Final simplification 76.6%

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

Alternative 10: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-180}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (<= t_0 5e-180) (* c0 (sqrt (/ (/ A l) V))) 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 <= 5e-180) {
		tmp = c0 * sqrt(((A / l) / V));
	} 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 <= 5d-180) then
        tmp = c0 * sqrt(((a / l) / v))
    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 <= 5e-180) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} 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 <= 5e-180:
		tmp = c0 * math.sqrt(((A / l) / V))
	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 <= 5e-180)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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 <= 5e-180)
		tmp = c0 * sqrt(((A / l) / V));
	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[LessEqual[t$95$0, 5e-180], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000001e-180

   1. Initial program 68.9%

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

      1. clear-num 68.2%

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

      2. associate-/r/ 68.9%

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

      3. associate-/r* 69.4%

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

   4. Applied egg-rr 69.4%

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

      1. associate-*l/ 72.4%

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

      2. associate-*r/ 70.2%

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

      3. associate-*l/ 70.2%

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

      4. *-un-lft-identity 70.2%

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

   6. Applied egg-rr 70.2%

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

   if 5.0000000000000001e-180 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 85.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 5 \cdot 10^{-180}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -8.8 \cdot 10^{-131}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -8.8e-131)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= A 1.3e-305)
     (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
     (* (sqrt A) (/ c0 (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -8.8e-131) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if (A <= 1.3e-305) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-8.8d-131)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if (a <= 1.3d-305) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else
        tmp = sqrt(a) * (c0 / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -8.8e-131) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if (A <= 1.3e-305) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -8.8e-131:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif A <= 1.3e-305:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -8.8e-131)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (A <= 1.3e-305)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -8.8e-131)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif (A <= 1.3e-305)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	else
		tmp = sqrt(A) * (c0 / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -8.8e-131], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.3e-305], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if A < -8.7999999999999999e-131

   1. Initial program 73.5%

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

      1. associate-/r* 74.9%

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

      2. sqrt-div 45.5%

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

      3. associate-*r/ 43.6%

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

   4. Applied egg-rr 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/l* 44.5%

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

      3. associate-/r/ 45.5%

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

   6. Simplified 45.5%

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

   if -8.7999999999999999e-131 < A < 1.3000000000000001e-305

   1. Initial program 82.2%

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

      1. clear-num 82.2%

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

      2. associate-/r/ 82.2%

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

      3. associate-/r* 82.3%

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

   4. Applied egg-rr 82.3%

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

   if 1.3000000000000001e-305 < A

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. sqrt-div 85.4%

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

      3. associate-*l/ 81.0%

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

   4. Applied egg-rr 81.0%

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

   6. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.25} \cdot \left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25} \cdot c0\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 72.4%

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

      2. associate-/l/ 69.3%

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

      3. associate-/l/ 72.5%

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

      4. pow-prod-up 72.6%

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

      5. metadata-eval 72.6%

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

      6. pow1/2 72.6%

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

      7. *-commutative 72.6%

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

      8. associate-/l/ 72.6%

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

      9. associate-/r* 72.7%

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

      10. sqrt-div 43.2%

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

      11. clear-num 43.2%

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

      12. div-inv 43.2%

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

      13. associate-/r/ 43.0%

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

      14. sqrt-div 51.5%

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

      15. frac-times 49.5%

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

      16. sqrt-prod 81.0%

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

      17. *-commutative 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. associate-/l* 85.3%

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

      2. associate-/r/ 84.6%

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

   10. Simplified 84.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -8.8 \cdot 10^{-131}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq 7 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \sqrt{\frac{A}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V 7e-309)
   (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
   (* (/ c0 (sqrt V)) (sqrt (/ A l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= 7e-309) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = (c0 / sqrt(V)) * sqrt((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) :: tmp
    if (v <= 7d-309) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else
        tmp = (c0 / sqrt(v)) * sqrt((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 tmp;
	if (V <= 7e-309) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = (c0 / Math.sqrt(V)) * Math.sqrt((A / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= 7e-309:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	else:
		tmp = (c0 / math.sqrt(V)) * math.sqrt((A / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= 7e-309)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(Float64(c0 / sqrt(V)) * sqrt(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)
	tmp = 0.0;
	if (V <= 7e-309)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	else
		tmp = (c0 / sqrt(V)) * sqrt((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_] := If[LessEqual[V, 7e-309], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < 6.9999999999999984e-309

   1. Initial program 72.6%

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

      1. clear-num 71.9%

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

      2. associate-/r/ 72.5%

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

      3. associate-/r* 72.6%

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

   4. Applied egg-rr 72.6%

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

   if 6.9999999999999984e-309 < V

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. sqrt-div 44.0%

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

      3. associate-*l/ 41.9%

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

   4. Applied egg-rr 41.9%

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

      1. *-commutative 41.9%

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

      2. sqrt-prod 47.9%

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

      3. times-frac 49.3%

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

      4. sqrt-div 83.4%

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

   6. Applied egg-rr 83.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq 7 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \sqrt{\frac{A}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% 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 74.3%

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

3. Final simplification 74.3%

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

Reproduce

herbie shell --seed 2024017 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))