Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.8% → 89.5%

Time: 7.6s

Alternatives: 16

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: 73.8% 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: 89.5% accurate, 0.5× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1e-310)
   (* (/ c0 (sqrt l)) (* (sqrt (/ -1.0 V)) (sqrt (- A))))
   (/ c0 (/ (sqrt (* l V)) (sqrt A)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-310) {
		tmp = (c0 / sqrt(l)) * (sqrt((-1.0 / V)) * sqrt(-A));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-1d-310)) then
        tmp = (c0 / sqrt(l)) * (sqrt(((-1.0d0) / v)) * sqrt(-a))
    else
        tmp = c0 / (sqrt((l * v)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-310) {
		tmp = (c0 / Math.sqrt(l)) * (Math.sqrt((-1.0 / V)) * Math.sqrt(-A));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -1e-310:
		tmp = (c0 / math.sqrt(l)) * (math.sqrt((-1.0 / V)) * math.sqrt(-A))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -1e-310)
		tmp = Float64(Float64(c0 / sqrt(l)) * Float64(sqrt(Float64(-1.0 / V)) * sqrt(Float64(-A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -1e-310)
		tmp = (c0 / sqrt(l)) * (sqrt((-1.0 / V)) * sqrt(-A));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -1e-310], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 73.0%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 47.0

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

   4. Applied rewrites 47.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. associate-/r/ N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 55.0

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

   6. Applied rewrites 55.0%

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

   if -9.999999999999969e-311 < A

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 86.4

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

   6. Applied rewrites 86.4%

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

4. Final simplification 70.0%

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

Alternative 2: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+252}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* l V))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+252) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+252) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (l * v)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+252) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (l * V)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+252) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (l * V)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+252:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+252], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000004e252

   1. Initial program 99.6%

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

   if 4.0000000000000004e252 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 64.9

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

   4. Applied rewrites 64.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 76.8

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

   6. Applied rewrites 76.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}} \leq 4 \cdot 10^{+252}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \left(-V\right)}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \sqrt{-\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -5e-213)
     (* c0 (* (sqrt (- A)) (/ 1.0 (sqrt (* l (- V))))))
     (if (<= (* l V) 0.0)
       (/ c0 (* (sqrt (- V)) (sqrt (- (/ l A)))))
       (/ c0 (/ (sqrt (* l V)) (sqrt A)))))))

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(l)) * sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt((l * -V))));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / (sqrt(-V) * sqrt(-(l / A)));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = c0 * (Math.sqrt(-A) * (1.0 / Math.sqrt((l * -V))));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / (Math.sqrt(-V) * Math.sqrt(-(l / A)));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	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(l)) * math.sqrt((A / V))
	elif (l * V) <= -5e-213:
		tmp = c0 * (math.sqrt(-A) * (1.0 / math.sqrt((l * -V))))
	elif (l * V) <= 0.0:
		tmp = c0 / (math.sqrt(-V) * math.sqrt(-(l / A)))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -5e-213)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * Float64(1.0 / sqrt(Float64(l * Float64(-V))))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(-V)) * sqrt(Float64(-Float64(l / A)))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -5e-213)
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt((l * -V))));
	elseif ((l * V) <= 0.0)
		tmp = c0 / (sqrt(-V) * sqrt(-(l / A)));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[(-N[(l / A), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 83.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. associate-*l/ N/A

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

      2. frac-2neg N/A

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

      3. *-lft-identity N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.2

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

      15. lift-*.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-neg.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 0.0

   1. Initial program 56.2%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 56.3

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

   4. Applied rewrites 56.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. sqrt-undiv N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. sqrt-undiv N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lift-neg.f64 N/A

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

      22. distribute-frac-neg2 N/A

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

      23. lower-neg.f64 N/A

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

      24. lower-/.f64 47.6

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

   6. Applied rewrites 47.6%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 77.9

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

   4. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 91.8

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

   6. Applied rewrites 91.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \left(-V\right)}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \sqrt{-\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \left(-V\right)}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -5e-213)
     (* c0 (* (sqrt (- A)) (/ 1.0 (sqrt (* l (- V))))))
     (if (<= (* l V) 0.0)
       (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
       (/ c0 (/ (sqrt (* l V)) (sqrt A)))))))

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(l)) * sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt((l * -V))));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = c0 * (Math.sqrt(-A) * (1.0 / Math.sqrt((l * -V))));
	} else if ((l * V) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	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(l)) * math.sqrt((A / V))
	elif (l * V) <= -5e-213:
		tmp = c0 * (math.sqrt(-A) * (1.0 / math.sqrt((l * -V))))
	elif (l * V) <= 0.0:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -5e-213)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * Float64(1.0 / sqrt(Float64(l * Float64(-V))))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -5e-213)
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt((l * -V))));
	elseif ((l * V) <= 0.0)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 83.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. associate-*l/ N/A

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

      2. frac-2neg N/A

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

      3. *-lft-identity N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.2

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

      15. lift-*.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-neg.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 0.0

   1. Initial program 56.2%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

      1. frac-2neg N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r* N/A

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

      4. un-div-inv N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 46.0

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

   6. Applied rewrites 46.0%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 77.9

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

   4. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 91.8

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

   6. Applied rewrites 91.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \left(-V\right)}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+287}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+287) (* 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 / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+287) {
		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 / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+287) 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 / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+287) {
		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 / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+287:
		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(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+287)
		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 / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+287)
		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[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+287], 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 37.8%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 52.4

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

   4. Applied rewrites 52.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.0000000000000001e287

   1. Initial program 98.5%

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

   if 1.0000000000000001e287 < (/.f64 A (*.f64 V l))

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 46.6

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

   4. Applied rewrites 46.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.7

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

   6. Applied rewrites 61.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 10^{+287}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+255}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2e+255) (* c0 (sqrt t_0)) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+255) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (l * v)
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+255) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+255) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+255:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+255)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+255)
		tmp = c0 * sqrt(t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+255], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 1.99999999999999998e255 < (/.f64 A (*.f64 V l))

   1. Initial program 44.9%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 58.3

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

   4. Applied rewrites 58.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999998e255

   1. Initial program 98.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+255}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \left(-V\right)}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -5e-213)
     (* c0 (* (sqrt (- A)) (/ 1.0 (sqrt (* l (- V))))))
     (if (<= (* l V) 5e-290)
       (/ c0 (sqrt (* l (/ V A))))
       (/ c0 (/ (sqrt (* l V)) (sqrt A)))))))

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(l)) * sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt((l * -V))));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = c0 * (Math.sqrt(-A) * (1.0 / Math.sqrt((l * -V))));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	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(l)) * math.sqrt((A / V))
	elif (l * V) <= -5e-213:
		tmp = c0 * (math.sqrt(-A) * (1.0 / math.sqrt((l * -V))))
	elif (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -5e-213)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) * Float64(1.0 / sqrt(Float64(l * Float64(-V))))));
	elseif (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -5e-213)
		tmp = c0 * (sqrt(-A) * (1.0 / sqrt((l * -V))));
	elseif ((l * V) <= 5e-290)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 83.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. associate-*l/ N/A

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

      2. frac-2neg N/A

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

      3. *-lft-identity N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.2

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

      15. lift-*.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-neg.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 77.4

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

   4. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   6. Applied rewrites 92.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;c0 \cdot \left(\sqrt{-A} \cdot \frac{1}{\sqrt{\ell \cdot \left(-V\right)}}\right)\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -5e-213)
     (/ (* c0 (sqrt (- A))) (sqrt (* l (- V))))
     (if (<= (* l V) 5e-290)
       (/ c0 (sqrt (* l (/ V A))))
       (/ c0 (/ (sqrt (* l V)) (sqrt A)))))))

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(l)) * sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	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(l)) * math.sqrt((A / V))
	elif (l * V) <= -5e-213:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt((l * -V))
	elif (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -5e-213)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(l * Float64(-V))));
	elseif (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -5e-213)
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	elseif ((l * V) <= 5e-290)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 83.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-prod N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 97.1%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 77.4

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

   4. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   6. Applied rewrites 92.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* l V) -5e-213)
     (/ (* c0 (sqrt (- A))) (sqrt (* l (- V))))
     (if (<= (* l V) 5e-290)
       (/ c0 (sqrt (* l (/ V A))))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	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(l)) * Math.sqrt((A / V));
	} else if ((l * V) <= -5e-213) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (l * V) <= -5e-213:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt((l * -V))
	elif (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(l * V) <= -5e-213)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(l * Float64(-V))));
	elseif (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((l * V) <= -5e-213)
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	elseif ((l * V) <= 5e-290)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 83.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-prod N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 97.1%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* l V) -5e-213)
     (/ (* c0 (sqrt (- A))) (sqrt (* l (- V))))
     (if (<= (* l V) 5e-290)
       (/ c0 (sqrt (* l (/ V A))))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((l * V) <= -5e-213) {
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	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-213) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (l * V) <= -5e-213:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt((l * -V))
	elif (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(l * V) <= -5e-213)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(l * Float64(-V))));
	elseif (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((l * V) <= -5e-213)
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	elseif ((l * V) <= 5e-290)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 83.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-prod N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 97.1%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e-213)
   (/ (* c0 (sqrt (- A))) (sqrt (* l (- V))))
   (if (<= (* l V) 5e-290)
     (/ c0 (sqrt (* l (/ V A))))
     (* c0 (/ (sqrt A) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e-213) {
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-5d-213)) then
        tmp = (c0 * sqrt(-a)) / sqrt((l * -v))
    else if ((l * v) <= 5d-290) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e-213) {
		tmp = (c0 * Math.sqrt(-A)) / Math.sqrt((l * -V));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e-213:
		tmp = (c0 * math.sqrt(-A)) / math.sqrt((l * -V))
	elif (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e-213)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / sqrt(Float64(l * Float64(-V))));
	elseif (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e-213)
		tmp = (c0 * sqrt(-A)) / sqrt((l * -V));
	elseif ((l * V) <= 5e-290)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 76.7%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 73.9

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

   4. Applied rewrites 73.9%

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

      1. lift-/.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-prod N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 87.8%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e-213)
   (* (sqrt (- A)) (/ c0 (sqrt (* l (- V)))))
   (if (<= (* l V) 5e-290)
     (/ c0 (sqrt (* l (/ V A))))
     (* c0 (/ (sqrt A) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e-213) {
		tmp = sqrt(-A) * (c0 / sqrt((l * -V)));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-5d-213)) then
        tmp = sqrt(-a) * (c0 / sqrt((l * -v)))
    else if ((l * v) <= 5d-290) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e-213) {
		tmp = Math.sqrt(-A) * (c0 / Math.sqrt((l * -V)));
	} else if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e-213:
		tmp = math.sqrt(-A) * (c0 / math.sqrt((l * -V)))
	elif (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e-213)
		tmp = Float64(sqrt(Float64(-A)) * Float64(c0 / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e-213)
		tmp = sqrt(-A) * (c0 / sqrt((l * -V)));
	elseif ((l * V) <= 5e-290)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e-213], N[(N[Sqrt[(-A)], $MachinePrecision] * N[(c0 / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999977e-213

   1. Initial program 76.7%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 73.9

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

   4. Applied rewrites 73.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 73.9

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

   6. Applied rewrites 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. frac-2neg N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. associate-/l/ N/A

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

      15. lift-*.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   8. Applied rewrites 87.7%

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

   if -4.99999999999999977e-213 < (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 59.9%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 73.0%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-sqrt.f64 N/A

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

      18. sqrt-prod N/A

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

      19. distribute-lft-neg-in N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 80.3%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 53.6

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

   8. Applied rewrites 53.6%

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

   if -9.999999999999969e-311 < A

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 86.4

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

   6. Applied rewrites 86.4%

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

4. Final simplification 69.2%

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

Alternative 14: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 5e-290)
   (/ c0 (sqrt (/ V (/ A l))))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 5e-290) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 5d-290) then
        tmp = c0 / sqrt((v / (a / l)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 5e-290) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 5e-290:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 5e-290)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 5e-290)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 5e-290], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 5.0000000000000001e-290

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 70.3

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

   4. Applied rewrites 70.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.8

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

   6. Applied rewrites 75.8%

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

   if 5.0000000000000001e-290 < (*.f64 V l)

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 10^{-301}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 1e-301)
   (/ c0 (sqrt (* V (/ l A))))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 1e-301) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 1d-301) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 1e-301) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 1e-301:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 1e-301)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 1e-301)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 1e-301], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 1.00000000000000007e-301

   1. Initial program 70.1%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 69.7

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

   4. Applied rewrites 69.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.2

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

   6. Applied rewrites 75.2%

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

   if 1.00000000000000007e-301 < (*.f64 V l)

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 92.4

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

   4. Applied rewrites 92.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 10^{-301}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Final simplification 73.1%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))