Henrywood and Agarwal, Equation (3)

Percentage Accurate: 72.6% → 89.3%

Time: 11.5s

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: 72.6% 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: 89.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $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 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-/r* 71.8%

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

      3. sqrt-div 44.3%

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

      4. associate-*l/ 42.9%

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

   4. Applied egg-rr 42.9%

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

      1. frac-2neg 42.9%

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

      2. sqrt-div 53.2%

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

   6. Applied egg-rr 53.2%

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

   if -4.999999999999985e-310 < A

   1. Initial program 75.5%

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

      1. pow1/2 75.5%

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

      2. div-inv 74.8%

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

      3. unpow-prod-down 86.3%

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

      4. pow1/2 86.3%

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

   4. Applied egg-rr 86.3%

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

      1. unpow1/2 86.3%

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

      2. associate-/r* 87.0%

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

   6. Simplified 87.0%

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

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \end{array} \]

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 (<= V -2e-311)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (/ (* c0 (sqrt A)) (* (sqrt l) (sqrt V)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2e-311) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else {
		tmp = (c0 * sqrt(A)) / (sqrt(l) * sqrt(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 (v <= (-2d-311)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else
        tmp = (c0 * sqrt(a)) / (sqrt(l) * sqrt(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 (V <= -2e-311) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else {
		tmp = (c0 * Math.sqrt(A)) / (Math.sqrt(l) * Math.sqrt(V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -2e-311:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	else:
		tmp = (c0 * math.sqrt(A)) / (math.sqrt(l) * math.sqrt(V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -2e-311)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / Float64(sqrt(l) * sqrt(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 (V <= -2e-311)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = (c0 * sqrt(A)) / (sqrt(l) * sqrt(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[V, -2e-311], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -1.9999999999999e-311

   1. Initial program 79.2%

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

      1. clear-num 78.3%

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

      2. associate-/r/ 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. associate-*l/ 79.2%

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

      2. *-un-lft-identity 79.2%

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

      3. associate-/l/ 76.9%

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

      4. frac-2neg 76.9%

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

      5. sqrt-div 86.5%

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

      6. distribute-neg-frac2 86.5%

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

   6. Applied egg-rr 86.5%

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

   if -1.9999999999999e-311 < V

   1. Initial program 72.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div 45.4%

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

      3. div-inv 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. *-commutative 45.3%

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

      2. associate-/r/ 45.3%

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

   6. Simplified 45.3%

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

      1. clear-num 45.3%

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

      2. inv-pow 45.3%

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

      3. sqrt-div 53.4%

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

      4. associate-/l/ 53.5%

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

      5. sqrt-prod 47.3%

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

      6. *-commutative 47.3%

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

      7. sqrt-div 72.6%

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

   8. Applied egg-rr 72.6%

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

      1. unpow-1 72.6%

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

   10. Simplified 72.6%

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

       1. associate-*r/ 72.6%

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

       2. *-un-lft-identity 72.6%

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

       3. times-frac 72.6%

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

       4. remove-double-div 72.7%

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

       5. associate-/r* 69.2%

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

       6. sqrt-div 45.4%

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

       7. times-frac 45.4%

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

       8. *-un-lft-identity 45.4%

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

       9. associate-*l/ 43.9%

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

       10. *-commutative 43.9%

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

       11. sqrt-div 53.5%

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

       12. frac-2neg 53.5%

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

       13. frac-times 55.7%

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

   12. Applied egg-rr 55.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= V -2e-311)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (/ (/ (* c0 (sqrt A)) (sqrt V)) (sqrt l))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2e-311) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else {
		tmp = ((c0 * sqrt(A)) / sqrt(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 (v <= (-2d-311)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else
        tmp = ((c0 * sqrt(a)) / sqrt(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 (V <= -2e-311) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else {
		tmp = ((c0 * Math.sqrt(A)) / Math.sqrt(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 V <= -2e-311:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	else:
		tmp = ((c0 * math.sqrt(A)) / math.sqrt(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 (V <= -2e-311)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(Float64(Float64(c0 * sqrt(A)) / sqrt(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 (V <= -2e-311)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = ((c0 * sqrt(A)) / sqrt(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[V, -2e-311], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -1.9999999999999e-311

   1. Initial program 79.2%

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

      1. clear-num 78.3%

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

      2. associate-/r/ 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. associate-*l/ 79.2%

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

      2. *-un-lft-identity 79.2%

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

      3. associate-/l/ 76.9%

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

      4. frac-2neg 76.9%

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

      5. sqrt-div 86.5%

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

      6. distribute-neg-frac2 86.5%

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

   6. Applied egg-rr 86.5%

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

   if -1.9999999999999e-311 < V

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/r* 69.2%

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

      3. sqrt-div 45.4%

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

      4. associate-*l/ 45.4%

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

   4. Applied egg-rr 45.4%

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

      1. sqrt-div 55.8%

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

      2. associate-*l/ 55.7%

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

   6. Applied egg-rr 55.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0 \cdot \sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{t\_0}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (sqrt l))))
   (if (<= (* V l) (- INFINITY))
     (/ t_0 (pow (/ A V) -0.5))
     (if (<= (* V l) -1e-300)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (* t_0 (sqrt (/ A 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 t_0 = c0 / sqrt(l);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0 / pow((A / V), -0.5);
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * sqrt((A / 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 t_0 = c0 / Math.sqrt(l);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 / Math.pow((A / V), -0.5);
	} else if ((V * l) <= -1e-300) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * Math.sqrt((A / 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):
	t_0 = c0 / math.sqrt(l)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0 / math.pow((A / V), -0.5)
	elif (V * l) <= -1e-300:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * math.sqrt((A / 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)
	t_0 = Float64(c0 / sqrt(l))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(t_0 / (Float64(A / V) ^ -0.5));
	elseif (Float64(V * l) <= -1e-300)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * sqrt(Float64(A / 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)
	t_0 = c0 / sqrt(l);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0 / ((A / V) ^ -0.5);
	elseif ((V * l) <= -1e-300)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * sqrt((A / 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_] := Block[{t$95$0 = N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(t$95$0 / N[Power[N[(A / V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-300], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[Sqrt[N[(A / 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 36.9%

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

      1. clear-num 36.9%

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

      2. associate-/r/ 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. associate-*l/ 36.9%

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

      2. *-un-lft-identity 36.9%

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

      3. associate-/r* 53.3%

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

      4. sqrt-undiv 42.6%

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

      5. div-inv 42.6%

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

      6. *-commutative 42.6%

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

      7. associate-/r/ 42.7%

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

      8. un-div-inv 42.8%

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

      9. div-inv 42.6%

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

      10. associate-/r* 42.7%

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

      11. pow1/2 42.7%

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

      12. pow-flip 42.7%

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

      13. metadata-eval 42.7%

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

   6. Applied egg-rr 42.7%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000003e-300

   1. Initial program 91.3%

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

      1. frac-2neg 91.3%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -1.00000000000000003e-300 < (*.f64 V l) < -0.0

   1. Initial program 36.6%

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

      1. associate-/r* 55.3%

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

      2. sqrt-div 34.5%

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

      3. associate-*r/ 34.5%

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

   4. Applied egg-rr 34.5%

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

      1. *-commutative 34.5%

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

      2. associate-/l* 34.6%

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

   6. Simplified 34.6%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 80.2%

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

      1. pow1/2 80.2%

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

      2. div-inv 79.5%

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

      3. unpow-prod-down 92.5%

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

      4. pow1/2 92.5%

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

   4. Applied egg-rr 92.5%

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

      1. unpow1/2 92.5%

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

      2. associate-/r* 93.3%

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

   6. Simplified 93.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;\frac{t\_0}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (sqrt l))))
   (if (<= (* V l) -4e+220)
     (/ t_0 (pow (/ A V) -0.5))
     (if (<= (* V l) -2e-275)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* t_0 (sqrt (/ A 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 t_0 = c0 / sqrt(l);
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = t_0 / pow((A / V), -0.5);
	} else if ((V * l) <= -2e-275) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * sqrt((A / 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) :: t_0
    real(8) :: tmp
    t_0 = c0 / sqrt(l)
    if ((v * l) <= (-4d+220)) then
        tmp = t_0 / ((a / v) ** (-0.5d0))
    else if ((v * l) <= (-2d-275)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 * sqrt((a / 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 t_0 = c0 / Math.sqrt(l);
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = t_0 / Math.pow((A / V), -0.5);
	} else if ((V * l) <= -2e-275) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * Math.sqrt((A / 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):
	t_0 = c0 / math.sqrt(l)
	tmp = 0
	if (V * l) <= -4e+220:
		tmp = t_0 / math.pow((A / V), -0.5)
	elif (V * l) <= -2e-275:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * math.sqrt((A / 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)
	t_0 = Float64(c0 / sqrt(l))
	tmp = 0.0
	if (Float64(V * l) <= -4e+220)
		tmp = Float64(t_0 / (Float64(A / V) ^ -0.5));
	elseif (Float64(V * l) <= -2e-275)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * sqrt(Float64(A / 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)
	t_0 = c0 / sqrt(l);
	tmp = 0.0;
	if ((V * l) <= -4e+220)
		tmp = t_0 / ((A / V) ^ -0.5);
	elseif ((V * l) <= -2e-275)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * sqrt((A / 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_] := Block[{t$95$0 = N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -4e+220], N[(t$95$0 / N[Power[N[(A / V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-275], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[Sqrt[N[(A / 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) < -4e220

   1. Initial program 44.1%

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

      1. clear-num 44.1%

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

      2. associate-/r/ 44.2%

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

   4. Applied egg-rr 44.2%

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

      1. associate-*l/ 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. associate-/r* 55.3%

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

      4. sqrt-undiv 50.0%

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

      5. div-inv 50.0%

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

      6. *-commutative 50.0%

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

      7. associate-/r/ 50.1%

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

      8. un-div-inv 50.3%

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

      9. div-inv 50.1%

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

      10. associate-/r* 50.2%

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

      11. pow1/2 50.2%

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

      12. pow-flip 50.3%

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

      13. metadata-eval 50.3%

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

   6. Applied egg-rr 50.3%

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

   if -4e220 < (*.f64 V l) < -1.99999999999999987e-275

   1. Initial program 94.9%

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

   if -1.99999999999999987e-275 < (*.f64 V l) < -0.0

   1. Initial program 39.6%

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

      1. associate-/r* 56.7%

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

      2. sqrt-div 37.2%

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

      3. associate-*r/ 37.2%

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

   4. Applied egg-rr 37.2%

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

      1. *-commutative 37.2%

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

      2. associate-/l* 37.3%

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

   6. Simplified 37.3%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 80.2%

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

      1. pow1/2 80.2%

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

      2. div-inv 79.5%

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

      3. unpow-prod-down 92.5%

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

      4. pow1/2 92.5%

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

   4. Applied egg-rr 92.5%

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

      1. unpow1/2 92.5%

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

      2. associate-/r* 93.3%

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

   6. Simplified 93.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;\frac{t\_0}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (sqrt l))))
   (if (<= (* V l) -4e+220)
     (/ t_0 (pow (/ A V) -0.5))
     (if (<= (* V l) -2e-275)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* t_0 (sqrt (/ A V)))
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / sqrt(l);
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = t_0 / pow((A / V), -0.5);
	} else if ((V * l) <= -2e-275) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * sqrt((A / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / sqrt(l)
    if ((v * l) <= (-4d+220)) then
        tmp = t_0 / ((a / v) ** (-0.5d0))
    else if ((v * l) <= (-2d-275)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 * sqrt((a / v))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / Math.sqrt(l);
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = t_0 / Math.pow((A / V), -0.5);
	} else if ((V * l) <= -2e-275) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * Math.sqrt((A / V));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / math.sqrt(l)
	tmp = 0
	if (V * l) <= -4e+220:
		tmp = t_0 / math.pow((A / V), -0.5)
	elif (V * l) <= -2e-275:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * math.sqrt((A / V))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / sqrt(l))
	tmp = 0.0
	if (Float64(V * l) <= -4e+220)
		tmp = Float64(t_0 / (Float64(A / V) ^ -0.5));
	elseif (Float64(V * l) <= -2e-275)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * sqrt(Float64(A / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / sqrt(l);
	tmp = 0.0;
	if ((V * l) <= -4e+220)
		tmp = t_0 / ((A / V) ^ -0.5);
	elseif ((V * l) <= -2e-275)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * sqrt((A / V));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -4e+220], N[(t$95$0 / N[Power[N[(A / V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-275], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4e220

   1. Initial program 44.1%

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

      1. clear-num 44.1%

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

      2. associate-/r/ 44.2%

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

   4. Applied egg-rr 44.2%

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

      1. associate-*l/ 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. associate-/r* 55.3%

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

      4. sqrt-undiv 50.0%

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

      5. div-inv 50.0%

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

      6. *-commutative 50.0%

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

      7. associate-/r/ 50.1%

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

      8. un-div-inv 50.3%

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

      9. div-inv 50.1%

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

      10. associate-/r* 50.2%

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

      11. pow1/2 50.2%

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

      12. pow-flip 50.3%

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

      13. metadata-eval 50.3%

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

   6. Applied egg-rr 50.3%

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

   if -4e220 < (*.f64 V l) < -1.99999999999999987e-275

   1. Initial program 94.9%

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

   if -1.99999999999999987e-275 < (*.f64 V l) < -0.0

   1. Initial program 39.6%

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

      1. associate-/r* 56.7%

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

      2. sqrt-div 37.2%

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

      3. associate-*r/ 37.2%

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

   4. Applied egg-rr 37.2%

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

      1. *-commutative 37.2%

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

      2. associate-/l* 37.3%

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

   6. Simplified 37.3%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 80.2%

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

      1. sqrt-div 93.2%

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

      2. associate-*r/ 91.7%

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

   4. Applied egg-rr 91.7%

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

      1. associate-/l* 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{{\left(\frac{A}{V}\right)}^{-0.5}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (/ c0 (sqrt l)) (sqrt (/ A V)))))
   (if (<= (* V l) -4e+220)
     t_0
     (if (<= (* V l) -2e-275)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / sqrt(l)) * sqrt((A / V));
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = t_0;
	} else if ((V * l) <= -2e-275) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / sqrt(l)) * sqrt((a / v))
    if ((v * l) <= (-4d+220)) then
        tmp = t_0
    else if ((v * l) <= (-2d-275)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = t_0;
	} else if ((V * l) <= -2e-275) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -4e+220:
		tmp = t_0
	elif (V * l) <= -2e-275:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)))
	tmp = 0.0
	if (Float64(V * l) <= -4e+220)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-275)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (c0 / sqrt(l)) * sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -4e+220)
		tmp = t_0;
	elseif ((V * l) <= -2e-275)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -4e+220], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-275], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4e220 or -1.99999999999999987e-275 < (*.f64 V l) < -0.0

   1. Initial program 41.3%

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

      1. associate-/r* 56.2%

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

      2. sqrt-div 42.2%

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

      3. associate-*r/ 42.2%

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

   4. Applied egg-rr 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-/l* 42.3%

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

   6. Simplified 42.3%

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

   if -4e220 < (*.f64 V l) < -1.99999999999999987e-275

   1. Initial program 94.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 80.2%

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

      1. sqrt-div 93.2%

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

      2. associate-*r/ 91.7%

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

   4. Applied egg-rr 91.7%

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

      1. associate-/l* 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -4e+138)
     t_0
     (if (<= (* V l) -5e-212)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (<= (* V l) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -4e+138) {
		tmp = t_0;
	} else if ((V * l) <= -5e-212) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-4d+138)) then
        tmp = t_0
    else if ((v * l) <= (-5d-212)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -4e+138) {
		tmp = t_0;
	} else if ((V * l) <= -5e-212) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -4e+138:
		tmp = t_0
	elif (V * l) <= -5e-212:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -4e+138)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-212)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -4e+138)
		tmp = t_0;
	elseif ((V * l) <= -5e-212)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -4e+138], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-212], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.0000000000000001e138 or -5.00000000000000043e-212 < (*.f64 V l) < -0.0

   1. Initial program 54.6%

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

      1. associate-/r* 65.8%

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

      2. sqrt-div 44.7%

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

      3. associate-*r/ 43.7%

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

   4. Applied egg-rr 43.7%

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

      1. associate-/l* 44.7%

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

   6. Simplified 44.7%

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

   if -4.0000000000000001e138 < (*.f64 V l) < -5.00000000000000043e-212

   1. Initial program 95.0%

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

      1. clear-num 95.0%

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

      2. associate-/r/ 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. associate-*l/ 95.0%

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

      2. *-un-lft-identity 95.0%

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

      3. associate-/r* 74.1%

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

      4. sqrt-undiv 42.0%

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

      5. div-inv 42.0%

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

      6. *-commutative 42.0%

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

      7. associate-/r/ 42.0%

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

      8. un-div-inv 42.1%

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

      9. sqrt-undiv 74.2%

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

      10. div-inv 74.1%

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

      11. clear-num 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. associate-*r/ 95.0%

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

      2. *-commutative 95.0%

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

      3. associate-/l* 84.8%

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

   8. Simplified 84.8%

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

   9. Taylor expanded in V around 0 95.0%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 80.2%

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

      1. sqrt-div 93.2%

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

      2. associate-*r/ 91.7%

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

   4. Applied egg-rr 91.7%

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

      1. associate-/l* 93.2%

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

   6. Simplified 93.2%

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

Alternative 9: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-166}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-166)
   (* c0 (sqrt (/ A (* V l))))
   (if (<= (* V l) 2e-310)
     (/ c0 (sqrt (* V (/ l A))))
     (* c0 (/ (sqrt A) (sqrt (* V l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-166) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 2e-310) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d-166)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 2d-310) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-166) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 2e-310) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-166:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 2e-310:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-166)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 2e-310)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-166)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 2e-310)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e-166], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-310], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.00000000000000004e-166

   1. Initial program 81.5%

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

   if -1.00000000000000004e-166 < (*.f64 V l) < 1.999999999999994e-310

   1. Initial program 57.4%

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

      1. clear-num 57.4%

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

      2. associate-/r/ 54.2%

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

   4. Applied egg-rr 54.2%

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

      1. associate-*l/ 57.4%

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

      2. *-un-lft-identity 57.4%

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

      3. associate-/r* 67.6%

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

      4. sqrt-undiv 40.0%

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

      5. div-inv 40.0%

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

      6. *-commutative 40.0%

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

      7. associate-/r/ 40.0%

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

      8. un-div-inv 40.0%

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

      9. sqrt-undiv 69.3%

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

      10. div-inv 69.3%

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

      11. clear-num 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. associate-*r/ 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-/l* 69.3%

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

   8. Simplified 69.3%

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

   if 1.999999999999994e-310 < (*.f64 V l)

   1. Initial program 80.0%

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

      1. sqrt-div 93.2%

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

      2. associate-*r/ 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. associate-/l* 93.2%

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

   6. Simplified 93.2%

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

Alternative 10: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+270}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+270)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+270)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * 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 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+270))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * 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 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+270)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+270):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * 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(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+270))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * 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 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+270)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * 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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+270]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 2.0000000000000001e270 < (/.f64 A (*.f64 V l))

   1. Initial program 43.1%

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

      1. *-commutative 43.1%

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

      2. associate-/l/ 52.8%

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

   3. Simplified 52.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e270

   1. Initial program 99.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+270}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+272) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+272) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+272) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+272) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+272:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+272)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+272)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $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+272], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-/l/ 50.9%

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

   3. Simplified 50.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e272

   1. Initial program 99.0%

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

   if 2.0000000000000001e272 < (/.f64 A (*.f64 V l))

   1. Initial program 44.8%

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

      1. clear-num 44.8%

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

      2. associate-/r/ 44.8%

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

   4. Applied egg-rr 44.8%

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

      1. associate-*l/ 44.8%

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

      2. *-un-lft-identity 44.8%

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

      3. associate-/r* 53.6%

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

      4. sqrt-undiv 38.3%

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

      5. div-inv 38.2%

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

      6. *-commutative 38.2%

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

      7. associate-/r/ 38.2%

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

      8. un-div-inv 38.2%

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

      9. sqrt-undiv 58.1%

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

      10. div-inv 58.1%

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

      11. clear-num 58.1%

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

   6. Applied egg-rr 58.1%

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

Alternative 12: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+297) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+297) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+297) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+297) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+297:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+297)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+297)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $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, 1e+297], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-/l/ 50.9%

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

   3. Simplified 50.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1e297

   1. Initial program 99.0%

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

   if 1e297 < (/.f64 A (*.f64 V l))

   1. Initial program 42.8%

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

      1. clear-num 42.8%

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

      2. associate-/r/ 42.8%

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

   4. Applied egg-rr 42.8%

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

      1. associate-*l/ 42.8%

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

      2. *-un-lft-identity 42.8%

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

      3. associate-/r* 53.6%

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

      4. sqrt-undiv 37.8%

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

      5. div-inv 37.8%

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

      6. *-commutative 37.8%

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

      7. associate-/r/ 37.8%

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

      8. un-div-inv 37.7%

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

      9. sqrt-undiv 58.4%

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

      10. div-inv 58.4%

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

      11. clear-num 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. associate-*r/ 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-/l* 57.0%

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

   8. Simplified 57.0%

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

Alternative 13: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))