Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 91.0%

Time: 9.6s

Alternatives: 14

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: 73.5% 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: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{{\ell}^{0.5}}{\frac{\sqrt{-A}}{\sqrt{-V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -4e-310)
   (/ c0 (/ (pow l 0.5) (/ (sqrt (- A)) (sqrt (- V)))))
   (/ c0 (/ (sqrt (* l V)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = c0 / (pow(l, 0.5) / (sqrt(-A) / sqrt(-V)));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-4d-310)) then
        tmp = c0 / ((l ** 0.5d0) / (sqrt(-a) / sqrt(-v)))
    else
        tmp = c0 / (sqrt((l * v)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = c0 / (Math.pow(l, 0.5) / (Math.sqrt(-A) / Math.sqrt(-V)));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -4e-310:
		tmp = c0 / (math.pow(l, 0.5) / (math.sqrt(-A) / math.sqrt(-V)))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -4e-310)
		tmp = Float64(c0 / Float64((l ^ 0.5) / Float64(sqrt(Float64(-A)) / sqrt(Float64(-V)))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -4e-310)
		tmp = c0 / ((l ^ 0.5) / (sqrt(-A) / sqrt(-V)));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -4e-310], N[(c0 / N[(N[Power[l, 0.5], $MachinePrecision] / N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $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 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 70.4%

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

      1. *-un-lft-identity 70.4%

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

      2. times-frac 70.2%

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

   3. Applied egg-rr 70.2%

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

      1. frac-times 70.4%

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

      2. *-un-lft-identity 70.4%

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

      3. associate-/r* 73.1%

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

      4. sqrt-undiv 38.0%

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

      5. associate-*r/ 38.0%

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

      6. add-sqr-sqrt 37.9%

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

      7. associate-/r* 37.9%

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

      8. pow1/2 37.9%

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

      9. sqrt-pow1 37.9%

         \[\leadsto \frac{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\color{blue}{{\ell}^{\left(\frac{0.5}{2}\right)}}}}{\sqrt{\sqrt{\ell}}} \]

      10. metadata-eval 37.9%

          \[\leadsto \frac{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{{\ell}^{\color{blue}{0.25}}}}{\sqrt{\sqrt{\ell}}} \]

      11. pow1/2 37.9%

          \[\leadsto \frac{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{{\ell}^{0.25}}}{\sqrt{\color{blue}{{\ell}^{0.5}}}} \]

      12. sqrt-pow1 37.9%

          \[\leadsto \frac{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{{\ell}^{0.25}}}{\color{blue}{{\ell}^{\left(\frac{0.5}{2}\right)}}} \]

      13. metadata-eval 37.9%

          \[\leadsto \frac{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{{\ell}^{0.25}}}{{\ell}^{\color{blue}{0.25}}} \]

   5. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{\frac{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{{\ell}^{0.25}}}{{\ell}^{0.25}}} \]
   6. Step-by-step derivation

      1. associate-/l/ 37.9%

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{{\ell}^{0.25} \cdot {\ell}^{0.25}}} \]

      2. associate-/l* 37.8%

         \[\leadsto \color{blue}{\frac{c0}{\frac{{\ell}^{0.25} \cdot {\ell}^{0.25}}{\sqrt{\frac{A}{V}}}}} \]

      3. pow-sqr 38.0%

         \[\leadsto \frac{c0}{\frac{\color{blue}{{\ell}^{\left(2 \cdot 0.25\right)}}}{\sqrt{\frac{A}{V}}}} \]

      4. metadata-eval 38.0%

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

   7. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{c0}{\frac{{\ell}^{0.5}}{\sqrt{\frac{A}{V}}}}} \]
   8. Step-by-step derivation

      1. frac-2neg 38.0%

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

      2. sqrt-div 42.1%

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

   9. Applied egg-rr 42.1%

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

   if -3.999999999999988e-310 < A

   1. Initial program 75.9%

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

      1. sqrt-div 83.8%

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

      2. associate-*r/ 83.0%

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

   3. Applied egg-rr 83.0%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{{\ell}^{0.5}}{\frac{\sqrt{-A}}{\sqrt{-V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]

Alternative 2: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -4e-310)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (/ c0 (/ (sqrt (* l V)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-4d-310)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else
        tmp = c0 / (sqrt((l * v)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -4e-310:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -4e-310)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -4e-310)
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	else
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -4e-310], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $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 < -3.999999999999988e-310

   1. Initial program 70.4%

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

      1. associate-/r* 73.1%

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

      2. sqrt-div 38.0%

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

   3. Applied egg-rr 38.0%

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

      1. frac-2neg 38.0%

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

      2. sqrt-div 42.1%

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

   5. Applied egg-rr 42.1%

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

   if -3.999999999999988e-310 < A

   1. Initial program 75.9%

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

      1. sqrt-div 83.8%

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

      2. associate-*r/ 83.0%

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

   3. Applied egg-rr 83.0%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]

Alternative 3: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 (or (<= t_0 0.0) (not (<= t_0 4e+301)))
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (/ c0 (pow t_0 -0.5)))))

C

assert(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) || !(t_0 <= 4e+301)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 / pow(t_0, -0.5);
	}
	return tmp;
}

Fortran

NOTE: 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) .or. (.not. (t_0 <= 4d+301))) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 / (t_0 ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert 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) || !(t_0 <= 4e+301)) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 / Math.pow(t_0, -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+301):
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 / math.pow(t_0, -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+301))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 / (t_0 ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+301)))
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 / (t_0 ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+301]], $MachinePrecision]], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.00000000000000021e301 < (/.f64 A (*.f64 V l))

   1. Initial program 33.2%

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

      1. associate-/r* 49.2%

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

      2. sqrt-div 42.1%

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

   3. Applied egg-rr 42.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000021e301

   1. Initial program 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. frac-times 98.4%

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

      2. *-un-lft-identity 98.4%

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

      3. associate-/r* 91.3%

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

      4. sqrt-undiv 47.4%

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

      5. clear-num 47.3%

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

      6. un-div-inv 47.3%

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

      7. sqrt-undiv 91.1%

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

      8. associate-/l* 97.8%

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

      9. *-commutative 97.8%

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

      10. *-un-lft-identity 97.8%

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

      11. times-frac 87.5%

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

      12. /-rgt-identity 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. *-commutative 87.5%

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

      2. associate-*l/ 97.8%

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

      3. associate-*r/ 90.7%

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

   7. Simplified 90.7%

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

      1. clear-num 90.5%

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

      2. div-inv 91.1%

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

      3. sqrt-undiv 47.3%

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

      4. clear-num 47.3%

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

      5. sqrt-div 91.3%

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

      6. pow1/2 91.3%

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

      7. pow-flip 91.3%

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

      8. associate-/l/ 98.4%

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

      9. metadata-eval 98.4%

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

   9. Applied egg-rr 98.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0 \lor \neg \left(\frac{A}{\ell \cdot V} \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{\ell \cdot V}\right)}^{-0.5}}\\ \end{array} \]

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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)) (sqrt l)))
     (if (<= t_0 1e+298)
       (/ c0 (pow t_0 -0.5))
       (/ c0 (* (sqrt l) (sqrt (/ V A))))))))

C

assert(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)) / sqrt(l));
	} else if (t_0 <= 1e+298) {
		tmp = c0 / pow(t_0, -0.5);
	} else {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: 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)) / sqrt(l))
    else if (t_0 <= 1d+298) then
        tmp = c0 / (t_0 ** (-0.5d0))
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert 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)) / Math.sqrt(l));
	} else if (t_0 <= 1e+298) {
		tmp = c0 / Math.pow(t_0, -0.5);
	} else {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([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)) / math.sqrt(l))
	elif t_0 <= 1e+298:
		tmp = c0 / math.pow(t_0, -0.5)
	else:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	return tmp

Julia

V, l = sort([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 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (t_0 <= 1e+298)
		tmp = Float64(c0 / (t_0 ^ -0.5));
	else
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([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)) / sqrt(l));
	elseif (t_0 <= 1e+298)
		tmp = c0 / (t_0 ^ -0.5);
	else
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(c0 / N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.0%

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

      1. associate-/r* 42.1%

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

      2. sqrt-div 37.9%

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

   3. Applied egg-rr 37.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. frac-times 98.4%

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

      2. *-un-lft-identity 98.4%

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

      3. associate-/r* 91.2%

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

      4. sqrt-undiv 47.0%

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

      5. clear-num 46.9%

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

      6. un-div-inv 47.0%

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

      7. sqrt-undiv 91.0%

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

      8. associate-/l* 97.8%

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

      9. *-commutative 97.8%

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

      10. *-un-lft-identity 97.8%

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

      11. times-frac 87.4%

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

      12. /-rgt-identity 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. *-commutative 87.4%

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

      2. associate-*l/ 97.8%

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

      3. associate-*r/ 90.7%

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

   7. Simplified 90.7%

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

      1. clear-num 90.4%

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

      2. div-inv 91.0%

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

      3. sqrt-undiv 47.0%

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

      4. clear-num 47.0%

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

      5. sqrt-div 91.2%

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

      6. pow1/2 91.2%

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

      7. pow-flip 91.3%

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

      8. associate-/l/ 98.4%

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

      9. metadata-eval 98.4%

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

   9. Applied egg-rr 98.4%

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

   if 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 38.7%

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

      1. *-un-lft-identity 38.7%

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

      2. times-frac 55.8%

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

   3. Applied egg-rr 55.8%

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

      1. frac-times 38.7%

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

      2. *-un-lft-identity 38.7%

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

      3. associate-/r* 55.7%

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

      4. sqrt-undiv 46.4%

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

      5. clear-num 46.4%

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

      6. un-div-inv 46.5%

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

      7. sqrt-undiv 58.0%

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

      8. associate-/l* 41.7%

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

      9. *-commutative 41.7%

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

      10. *-un-lft-identity 41.7%

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

      11. times-frac 58.8%

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

      12. /-rgt-identity 58.8%

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

   5. Applied egg-rr 58.8%

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

      1. *-commutative 58.8%

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

      2. associate-*l/ 41.7%

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

      3. associate-*r/ 58.0%

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

   7. Simplified 58.0%

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

      1. *-commutative 58.0%

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

      2. sqrt-prod 46.5%

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

   9. Applied egg-rr 46.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 10^{+298}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{\ell \cdot V}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]

Alternative 5: 84.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / l) / v)))
    else
        tmp = c0 / (sqrt((v / a)) / (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] / N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.8%

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

      1. *-un-lft-identity 69.8%

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

      2. times-frac 70.8%

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

   3. Applied egg-rr 70.8%

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

      1. frac-times 69.8%

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

      2. *-un-lft-identity 69.8%

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

      3. associate-/r* 71.4%

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

   5. Applied egg-rr 71.4%

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

      1. associate-/l/ 69.8%

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

      2. sqrt-div 31.6%

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

      3. sqrt-prod 0.0%

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

      4. /-rgt-identity 0.0%

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

      5. sqrt-prod 31.6%

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

      6. associate-/r/ 31.6%

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

      7. sqrt-div 69.8%

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

      8. div-inv 69.3%

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

      9. sqrt-prod 31.6%

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

      10. associate-/r/ 31.6%

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

      11. /-rgt-identity 31.6%

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

   7. Applied egg-rr 31.6%

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

      1. associate-/r* 32.3%

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

   9. Simplified 32.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 75.0%

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

   3. Applied egg-rr 75.0%

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

      1. frac-times 76.0%

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

      2. *-un-lft-identity 76.0%

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

      3. associate-/r* 78.2%

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

      4. sqrt-undiv 89.9%

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

      5. clear-num 89.7%

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

      6. un-div-inv 89.9%

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

      7. sqrt-undiv 78.5%

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

      8. associate-/l* 75.9%

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

      9. *-commutative 75.9%

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

      10. *-un-lft-identity 75.9%

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

      11. times-frac 75.0%

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

      12. /-rgt-identity 75.0%

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

   5. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*l/ 75.9%

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

      3. associate-*r/ 78.5%

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

   7. Simplified 78.5%

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

      1. *-un-lft-identity 78.5%

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

      2. sqrt-prod 89.9%

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

      3. times-frac 89.9%

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

      4. pow1/2 89.9%

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

      5. pow-flip 90.0%

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

      6. metadata-eval 90.0%

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

   9. Applied egg-rr 90.0%

      \[\leadsto \color{blue}{{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}} \]
   10. Step-by-step derivation

       1. *-commutative 90.0%

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

       2. associate-*l/ 88.0%

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

       3. associate-/l* 90.0%

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

   11. Simplified 90.0%

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

4. Final simplification 61.4%

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

Alternative 6: 84.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / l) / v)))
    else
        tmp = c0 * (((v / a) ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.8%

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

      1. *-un-lft-identity 69.8%

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

      2. times-frac 70.8%

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

   3. Applied egg-rr 70.8%

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

      1. frac-times 69.8%

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

      2. *-un-lft-identity 69.8%

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

      3. associate-/r* 71.4%

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

   5. Applied egg-rr 71.4%

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

      1. associate-/l/ 69.8%

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

      2. sqrt-div 31.6%

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

      3. sqrt-prod 0.0%

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

      4. /-rgt-identity 0.0%

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

      5. sqrt-prod 31.6%

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

      6. associate-/r/ 31.6%

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

      7. sqrt-div 69.8%

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

      8. div-inv 69.3%

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

      9. sqrt-prod 31.6%

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

      10. associate-/r/ 31.6%

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

      11. /-rgt-identity 31.6%

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

   7. Applied egg-rr 31.6%

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

      1. associate-/r* 32.3%

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

   9. Simplified 32.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 75.0%

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

   3. Applied egg-rr 75.0%

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

      1. frac-times 76.0%

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

      2. associate-/l* 75.5%

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

      3. inv-pow 75.5%

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

      4. associate-/l* 74.5%

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

      5. metadata-eval 74.5%

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

      6. pow-prod-up 74.5%

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

      7. sqrt-unprod 74.6%

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

      8. add-sqr-sqrt 74.8%

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

      9. associate-/r/ 78.5%

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

      10. unpow-prod-down 90.0%

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

   5. Applied egg-rr 90.0%

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

4. Final simplification 61.4%

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

Alternative 7: 84.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = c0 / (sqrt((l * v)) / sqrt(a))
    else
        tmp = c0 / (sqrt((v / a)) / (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] / N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.8%

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

      1. sqrt-div 31.6%

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

      2. associate-*r/ 30.9%

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

   3. Applied egg-rr 30.9%

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

      1. associate-/l* 31.6%

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

   5. Simplified 31.6%

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

   if -4.999999999999985e-310 < l

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 75.0%

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

   3. Applied egg-rr 75.0%

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

      1. frac-times 76.0%

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

      2. *-un-lft-identity 76.0%

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

      3. associate-/r* 78.2%

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

      4. sqrt-undiv 89.9%

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

      5. clear-num 89.7%

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

      6. un-div-inv 89.9%

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

      7. sqrt-undiv 78.5%

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

      8. associate-/l* 75.9%

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

      9. *-commutative 75.9%

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

      10. *-un-lft-identity 75.9%

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

      11. times-frac 75.0%

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

      12. /-rgt-identity 75.0%

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

   5. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*l/ 75.9%

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

      3. associate-*r/ 78.5%

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

   7. Simplified 78.5%

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

      1. *-un-lft-identity 78.5%

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

      2. sqrt-prod 89.9%

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

      3. times-frac 89.9%

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

      4. pow1/2 89.9%

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

      5. pow-flip 90.0%

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

      6. metadata-eval 90.0%

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

   9. Applied egg-rr 90.0%

      \[\leadsto \color{blue}{{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}} \]
   10. Step-by-step derivation

       1. *-commutative 90.0%

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

       2. associate-*l/ 88.0%

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

       3. associate-/l* 90.0%

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

   11. Simplified 90.0%

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

4. Final simplification 61.0%

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

Alternative 8: 84.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.8%

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

      1. sqrt-div 31.6%

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

      2. associate-*r/ 30.9%

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

   3. Applied egg-rr 30.9%

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

      1. associate-/l* 31.6%

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

   5. Simplified 31.6%

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

   if -4.999999999999985e-310 < l

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 75.0%

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

   3. Applied egg-rr 75.0%

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

      1. frac-times 76.0%

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

      2. *-un-lft-identity 76.0%

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

      3. associate-/r* 78.2%

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

      4. sqrt-undiv 89.9%

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

      5. clear-num 89.7%

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

      6. un-div-inv 89.9%

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

      7. sqrt-undiv 78.5%

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

      8. associate-/l* 75.9%

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

      9. *-commutative 75.9%

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

      10. *-un-lft-identity 75.9%

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

      11. times-frac 75.0%

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

      12. /-rgt-identity 75.0%

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

   5. Applied egg-rr 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*l/ 75.9%

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

      3. associate-*r/ 78.5%

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

   7. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. sqrt-prod 89.9%

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

   9. Applied egg-rr 89.9%

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

4. Final simplification 61.0%

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

Alternative 9: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+283}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 l) V)))
     (if (<= t_0 1e+283) (/ c0 (pow t_0 -0.5)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(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 / l) / V));
	} else if (t_0 <= 1e+283) {
		tmp = c0 / pow(t_0, -0.5);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 / l) / v))
    else if (t_0 <= 1d+283) then
        tmp = c0 / (t_0 ** (-0.5d0))
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert 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 / l) / V));
	} else if (t_0 <= 1e+283) {
		tmp = c0 / Math.pow(t_0, -0.5);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([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 / l) / V))
	elif t_0 <= 1e+283:
		tmp = c0 / math.pow(t_0, -0.5)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([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 / l) / V)));
	elseif (t_0 <= 1e+283)
		tmp = Float64(c0 / (t_0 ^ -0.5));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([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 / l) / V));
	elseif (t_0 <= 1e+283)
		tmp = c0 / (t_0 ^ -0.5);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], N[(c0 / N[Power[t$95$0, -0.5], $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 28.0%

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

      1. *-un-lft-identity 28.0%

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

      2. times-frac 42.1%

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

   3. Applied egg-rr 42.1%

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

      1. associate-*l/ 42.1%

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

      2. *-un-lft-identity 42.1%

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

   5. Applied egg-rr 42.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.99999999999999955e282

   1. Initial program 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. times-frac 87.7%

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

   3. Applied egg-rr 87.7%

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

      1. frac-times 98.4%

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

      2. *-un-lft-identity 98.4%

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

      3. associate-/r* 91.0%

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

      4. sqrt-undiv 46.6%

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

      5. clear-num 46.5%

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

      6. un-div-inv 46.6%

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

      7. sqrt-undiv 90.7%

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

      8. associate-/l* 97.7%

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

      9. *-commutative 97.7%

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

      10. *-un-lft-identity 97.7%

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

      11. times-frac 87.0%

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

      12. /-rgt-identity 87.0%

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

   5. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*l/ 97.7%

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

      3. associate-*r/ 90.3%

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

   7. Simplified 90.3%

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

      1. clear-num 90.1%

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

      2. div-inv 90.7%

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

      3. sqrt-undiv 46.6%

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

      4. clear-num 46.6%

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

      5. sqrt-div 91.0%

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

      6. pow1/2 90.9%

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

      7. pow-flip 91.0%

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

      8. associate-/l/ 98.4%

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

      9. metadata-eval 98.4%

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

   9. Applied egg-rr 98.4%

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

   if 9.99999999999999955e282 < (/.f64 A (*.f64 V l))

   1. Initial program 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. times-frac 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. frac-times 43.7%

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

      2. *-un-lft-identity 43.7%

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

      3. associate-/r* 59.3%

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

      4. sqrt-undiv 47.5%

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

      5. clear-num 47.5%

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

      6. un-div-inv 47.6%

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

      7. sqrt-undiv 61.4%

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

      8. associate-/l* 46.5%

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

      9. *-commutative 46.5%

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

      10. *-un-lft-identity 46.5%

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

      11. times-frac 62.1%

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

      12. /-rgt-identity 62.1%

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

   5. Applied egg-rr 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l/ 46.5%

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

      3. associate-*r/ 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 10^{+283}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{\ell \cdot V}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 10: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 (or (<= t_0 0.0) (not (<= t_0 4e+301)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(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) || !(t_0 <= 4e+301)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: 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) .or. (.not. (t_0 <= 4d+301))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert 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) || !(t_0 <= 4e+301)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+301):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+301))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+301)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+301]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.00000000000000021e301 < (/.f64 A (*.f64 V l))

   1. Initial program 33.2%

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

      1. *-un-lft-identity 33.2%

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

      2. times-frac 49.2%

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

   3. Applied egg-rr 49.2%

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

      1. frac-times 33.2%

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

      2. *-un-lft-identity 33.2%

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

      3. associate-/r* 49.2%

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

   5. Applied egg-rr 49.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000021e301

   1. Initial program 98.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0 \lor \neg \left(\frac{A}{\ell \cdot V} \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]

Alternative 11: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 (or (<= t_0 0.0) (not (<= t_0 4e+301)))
     (* c0 (sqrt (/ (/ A l) V)))
     (* c0 (sqrt t_0)))))

C

assert(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) || !(t_0 <= 4e+301)) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: 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) .or. (.not. (t_0 <= 4d+301))) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert 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) || !(t_0 <= 4e+301)) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+301):
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+301))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+301)))
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+301]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.00000000000000021e301 < (/.f64 A (*.f64 V l))

   1. Initial program 33.2%

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

      1. *-un-lft-identity 33.2%

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

      2. times-frac 49.2%

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

   3. Applied egg-rr 49.2%

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

      1. associate-*l/ 49.2%

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

      2. *-un-lft-identity 49.2%

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

   5. Applied egg-rr 49.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000021e301

   1. Initial program 98.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0 \lor \neg \left(\frac{A}{\ell \cdot V} \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]

Alternative 12: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 l) V)))
     (if (<= t_0 5e+274) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(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 / l) / V));
	} else if (t_0 <= 5e+274) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 / l) / v))
    else if (t_0 <= 5d+274) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert 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 / l) / V));
	} else if (t_0 <= 5e+274) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([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 / l) / V))
	elif t_0 <= 5e+274:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([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 / l) / V)));
	elseif (t_0 <= 5e+274)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([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 / l) / V));
	elseif (t_0 <= 5e+274)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+274], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.0%

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

      1. *-un-lft-identity 28.0%

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

      2. times-frac 42.1%

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

   3. Applied egg-rr 42.1%

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

      1. associate-*l/ 42.1%

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

      2. *-un-lft-identity 42.1%

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

   5. Applied egg-rr 42.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999998e274

   1. Initial program 98.3%

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

   if 4.9999999999999998e274 < (/.f64 A (*.f64 V l))

   1. Initial program 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. times-frac 60.0%

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

   3. Applied egg-rr 60.0%

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

      1. frac-times 44.6%

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

      2. *-un-lft-identity 44.6%

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

      3. associate-/r* 58.5%

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

      4. sqrt-undiv 46.9%

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

      5. clear-num 46.9%

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

      6. un-div-inv 47.0%

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

      7. sqrt-undiv 60.5%

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

      8. associate-/l* 47.3%

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

      9. *-commutative 47.3%

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

      10. *-un-lft-identity 47.3%

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

      11. times-frac 62.7%

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

      12. /-rgt-identity 62.7%

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

   5. Applied egg-rr 62.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 13: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+283}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 l) V)))
     (if (<= t_0 1e+283) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(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 / l) / V));
	} else if (t_0 <= 1e+283) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 / l) / v))
    else if (t_0 <= 1d+283) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert 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 / l) / V));
	} else if (t_0 <= 1e+283) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([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 / l) / V))
	elif t_0 <= 1e+283:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([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 / l) / V)));
	elseif (t_0 <= 1e+283)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([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 / l) / V));
	elseif (t_0 <= 1e+283)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+283], 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 28.0%

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

      1. *-un-lft-identity 28.0%

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

      2. times-frac 42.1%

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

   3. Applied egg-rr 42.1%

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

      1. associate-*l/ 42.1%

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

      2. *-un-lft-identity 42.1%

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

   5. Applied egg-rr 42.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.99999999999999955e282

   1. Initial program 98.4%

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

   if 9.99999999999999955e282 < (/.f64 A (*.f64 V l))

   1. Initial program 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. times-frac 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. frac-times 43.7%

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

      2. *-un-lft-identity 43.7%

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

      3. associate-/r* 59.3%

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

      4. sqrt-undiv 47.5%

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

      5. clear-num 47.5%

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

      6. un-div-inv 47.6%

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

      7. sqrt-undiv 61.4%

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

      8. associate-/l* 46.5%

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

      9. *-commutative 46.5%

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

      10. *-un-lft-identity 46.5%

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

      11. times-frac 62.1%

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

      12. /-rgt-identity 62.1%

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

   5. Applied egg-rr 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l/ 46.5%

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

      3. associate-*r/ 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 10^{+283}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 14: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: 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 73.0%

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

2. Final simplification 73.0%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))