Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 90.9%

Time: 12.4s

Alternatives: 13

Speedup: 1.0×

• Rival profile vector overflowed, profile may not be complete

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: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (* (sqrt (/ A V)) (pow l -0.5)))
   (if (<= (* V l) -2e-287)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (/ c0 (* (sqrt (/ V A)) (sqrt l)))
       (if (<= (* V l) 1e+305)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* (sqrt A) (/ (/ c0 (sqrt l)) (sqrt V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) * pow(l, -0.5));
	} else if ((V * l) <= -2e-287) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(A) * ((c0 / sqrt(l)) / sqrt(V));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.pow(l, -0.5));
	} else if ((V * l) <= -2e-287) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(A) * ((c0 / Math.sqrt(l)) / Math.sqrt(V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) * math.pow(l, -0.5))
	elif (V * l) <= -2e-287:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(A) * ((c0 / math.sqrt(l)) / math.sqrt(V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * (l ^ -0.5)));
	elseif (Float64(V * l) <= -2e-287)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(sqrt(A) * Float64(Float64(c0 / sqrt(l)) / sqrt(V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) * (l ^ -0.5));
	elseif ((V * l) <= -2e-287)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(A) * ((c0 / sqrt(l)) / sqrt(V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-287], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 43.2%

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

      1. associate-/r* 84.6%

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

      2. div-inv 84.6%

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

      3. div-inv 84.6%

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

      4. associate-*l* 45.6%

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

   4. Applied egg-rr 45.6%

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

      1. associate-*r* 84.6%

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

      2. div-inv 84.6%

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

      3. sqrt-prod 66.7%

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

      4. inv-pow 66.7%

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

      5. sqrt-pow1 66.7%

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

      6. metadata-eval 66.7%

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

   6. Applied egg-rr 66.7%

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

   if -inf.0 < (*.f64 V l) < -2.00000000000000004e-287

   1. Initial program 77.9%

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

      1. frac-2neg 77.9%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -2.00000000000000004e-287 < (*.f64 V l) < -0.0

   1. Initial program 39.1%

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

      1. associate-/r* 66.5%

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

      2. div-inv 66.5%

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

      3. div-inv 66.5%

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

      4. associate-*l* 38.3%

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

   4. Applied egg-rr 38.3%

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

      1. frac-times 38.3%

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

      2. metadata-eval 38.3%

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

      3. div-inv 39.1%

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

      4. sqrt-undiv 10.3%

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

      5. clear-num 10.3%

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

      6. un-div-inv 10.3%

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

      7. sqrt-undiv 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/l* 66.5%

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

   8. Simplified 66.5%

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

      1. *-commutative 66.5%

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

      2. sqrt-prod 50.1%

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

   10. Applied egg-rr 50.1%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.6%

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

      1. sqrt-div 99.0%

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

      2. associate-*r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 15.9%

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

      1. frac-2neg 15.9%

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

      2. sqrt-div 0.0%

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

      3. *-commutative 0.0%

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

      4. distribute-rgt-neg-in 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sqrt-prod 0.0%

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

      3. times-frac 0.0%

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

      4. sqrt-div 75.2%

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

      5. frac-2neg 75.2%

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

      6. *-commutative 75.2%

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

      7. sqrt-div 87.0%

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

      8. associate-*l/ 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. associate-/l* 87.3%

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

   8. Simplified 87.3%

      \[\leadsto \color{blue}{\sqrt{A} \cdot \frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{V}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-171} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+266}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 5e-171) (not (<= t_0 5e+266)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 5e-171) || !(t_0 <= 5e+266)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 5d-171) .or. (.not. (t_0 <= 5d+266))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 5e-171) || !(t_0 <= 5e+266)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 5e-171) or not (t_0 <= 5e+266):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 5e-171) || !(t_0 <= 5e+266))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 5e-171) || ~((t_0 <= 5e+266)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-171], N[Not[LessEqual[t$95$0, 5e+266]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.99999999999999992e-171 or 4.9999999999999999e266 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/l/ 69.8%

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

   3. Simplified 69.8%

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

   if 4.99999999999999992e-171 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999999e266

   1. Initial program 98.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-171} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+266}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-235)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 1e+285)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-235) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 1e+285) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-235) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 1d+285) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-235) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 1e+285) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-235:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 1e+285:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-235)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 1e+285)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-235)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 1e+285)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-235], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+285], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $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 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999998e-235

   1. Initial program 67.3%

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

   3. Taylor expanded in c0 around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/r* 73.1%

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

   5. Simplified 73.1%

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

   if 4.9999999999999998e-235 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999998e284

   1. Initial program 98.6%

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

      1. associate-/r* 85.3%

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

      2. div-inv 85.3%

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

      3. div-inv 85.2%

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

      4. associate-*l* 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. frac-times 98.1%

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

      2. metadata-eval 98.1%

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

      3. div-inv 98.6%

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

      4. sqrt-undiv 44.3%

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

      5. clear-num 44.3%

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

      6. un-div-inv 44.2%

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

      7. sqrt-undiv 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 9.9999999999999998e284 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 47.3%

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

      1. associate-/r* 55.3%

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

      2. div-inv 55.3%

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

      3. div-inv 55.3%

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

      4. associate-*l* 47.3%

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

   4. Applied egg-rr 47.3%

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

      1. frac-times 47.3%

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

      2. metadata-eval 47.3%

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

      3. div-inv 47.3%

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

      4. sqrt-undiv 33.7%

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

      5. clear-num 33.7%

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

      6. un-div-inv 33.7%

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

      7. sqrt-undiv 53.3%

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

   6. Applied egg-rr 53.3%

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

      1. associate-*r/ 57.3%

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

   8. Simplified 57.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+285}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+260}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-261)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 4e+260) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-261) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+260) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-261) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 4d+260) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-261) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+260) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-261:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+260:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-261)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+260)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-261)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+260)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-261], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+260], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.99999999999999981e-261

   1. Initial program 67.3%

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

   3. Taylor expanded in c0 around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/r* 73.1%

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

   5. Simplified 73.1%

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

   if 4.99999999999999981e-261 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.00000000000000026e260

   1. Initial program 98.6%

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

   if 4.00000000000000026e260 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 51.3%

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

      1. associate-/r* 54.0%

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

      2. div-inv 54.0%

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

      3. div-inv 54.0%

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

      4. associate-*l* 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. frac-times 51.3%

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

      2. metadata-eval 51.3%

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

      3. div-inv 51.3%

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

      4. sqrt-undiv 38.8%

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

      5. clear-num 38.8%

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

      6. un-div-inv 38.8%

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

      7. sqrt-undiv 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. associate-*r/ 60.6%

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

   8. Simplified 60.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+260}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 5e-261)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+266) t_0 (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-261) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 5d-261) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+266) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 5e-261) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 5e-261:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+266:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 5e-261)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+266)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 5e-261)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+266)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-261], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+266], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.99999999999999981e-261

   1. Initial program 67.3%

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

   3. Taylor expanded in c0 around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/r* 73.1%

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

   5. Simplified 73.1%

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

   if 4.99999999999999981e-261 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999999e266

   1. Initial program 98.6%

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

   if 4.9999999999999999e266 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

      2. associate-/l/ 56.5%

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

   3. Simplified 56.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+266}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{V}{A}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{t\_0 \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot {\ell}^{-0.5}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ V A))))
   (if (<= (* V l) (- INFINITY))
     (* c0 (* (sqrt (/ A V)) (pow l -0.5)))
     (if (<= (* V l) -2e-287)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (/ c0 (* t_0 (sqrt l)))
         (if (<= (* V l) 1e+305)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ (* c0 (pow l -0.5)) t_0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((V / A));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) * pow(l, -0.5));
	} else if ((V * l) <= -2e-287) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (t_0 * sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0 * pow(l, -0.5)) / t_0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((V / A));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.pow(l, -0.5));
	} else if ((V * l) <= -2e-287) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (t_0 * Math.sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0 * Math.pow(l, -0.5)) / t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((V / A))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) * math.pow(l, -0.5))
	elif (V * l) <= -2e-287:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / (t_0 * math.sqrt(l))
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0 * math.pow(l, -0.5)) / t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(V / A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * (l ^ -0.5)));
	elseif (Float64(V * l) <= -2e-287)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(t_0 * sqrt(l)));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0 * (l ^ -0.5)) / t_0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((V / A));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) * (l ^ -0.5));
	elseif ((V * l) <= -2e-287)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (t_0 * sqrt(l));
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0 * (l ^ -0.5)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-287], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(t$95$0 * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 43.2%

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

      1. associate-/r* 84.6%

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

      2. div-inv 84.6%

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

      3. div-inv 84.6%

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

      4. associate-*l* 45.6%

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

   4. Applied egg-rr 45.6%

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

      1. associate-*r* 84.6%

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

      2. div-inv 84.6%

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

      3. sqrt-prod 66.7%

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

      4. inv-pow 66.7%

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

      5. sqrt-pow1 66.7%

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

      6. metadata-eval 66.7%

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

   6. Applied egg-rr 66.7%

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

   if -inf.0 < (*.f64 V l) < -2.00000000000000004e-287

   1. Initial program 77.9%

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

      1. frac-2neg 77.9%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -2.00000000000000004e-287 < (*.f64 V l) < -0.0

   1. Initial program 39.1%

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

      1. associate-/r* 66.5%

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

      2. div-inv 66.5%

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

      3. div-inv 66.5%

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

      4. associate-*l* 38.3%

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

   4. Applied egg-rr 38.3%

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

      1. frac-times 38.3%

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

      2. metadata-eval 38.3%

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

      3. div-inv 39.1%

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

      4. sqrt-undiv 10.3%

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

      5. clear-num 10.3%

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

      6. un-div-inv 10.3%

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

      7. sqrt-undiv 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/l* 66.5%

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

   8. Simplified 66.5%

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

      1. *-commutative 66.5%

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

      2. sqrt-prod 50.1%

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

   10. Applied egg-rr 50.1%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.6%

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

      1. sqrt-div 99.0%

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

      2. associate-*r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 15.9%

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

      1. associate-/r* 68.5%

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

      2. div-inv 68.5%

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

      3. div-inv 68.5%

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

      4. associate-*l* 27.5%

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

   4. Applied egg-rr 27.5%

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

      1. frac-times 15.9%

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

      2. metadata-eval 15.9%

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

      3. div-inv 15.9%

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

      4. sqrt-undiv 15.9%

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

      5. clear-num 15.9%

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

      6. un-div-inv 15.9%

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

      7. sqrt-undiv 15.9%

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

   6. Applied egg-rr 15.9%

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

      1. *-commutative 15.9%

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

      2. associate-/l* 64.0%

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

   8. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. sqrt-prod 75.3%

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

      3. times-frac 74.9%

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

      4. add-sqr-sqrt 74.9%

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

      5. sqrt-unprod 27.0%

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

      6. sqr-neg 27.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. pow1/2 0.0%

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

      10. pow-flip 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 27.0%

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

      13. sqr-neg 27.0%

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

      14. sqrt-unprod 75.1%

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

      15. add-sqr-sqrt 75.3%

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

      16. metadata-eval 75.3%

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

   10. Applied egg-rr 75.3%

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

       1. associate-*r/ 75.5%

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

       2. *-commutative 75.5%

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

   12. Simplified 75.5%

       \[\leadsto \color{blue}{\frac{c0 \cdot {\ell}^{-0.5}}{\sqrt{\frac{V}{A}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot {\ell}^{-0.5}}{\sqrt{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+305}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A V)) (/ c0 (sqrt l)))))
   (if (<= (* V l) -5e+177)
     t_0
     (if (<= (* V l) -5e-121)
       (/ c0 (sqrt (/ (* V l) A)))
       (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+305)))
         t_0
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V)) * (c0 / sqrt(l));
	double tmp;
	if ((V * l) <= -5e+177) {
		tmp = t_0;
	} else if ((V * l) <= -5e-121) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+305)) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v)) * (c0 / sqrt(l))
    if ((v * l) <= (-5d+177)) then
        tmp = t_0
    else if ((v * l) <= (-5d-121)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+305))) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -5e+177) {
		tmp = t_0;
	} else if ((V * l) <= -5e-121) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+305)) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	tmp = 0
	if (V * l) <= -5e+177:
		tmp = t_0
	elif (V * l) <= -5e-121:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+305):
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+177)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-121)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+305))
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V)) * (c0 / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -5e+177)
		tmp = t_0;
	elseif ((V * l) <= -5e-121)
		tmp = c0 / sqrt(((V * l) / A));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+305)))
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+177], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-121], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+305]], $MachinePrecision]], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -5.0000000000000003e177 or -4.99999999999999989e-121 < (*.f64 V l) < -0.0 or 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 47.6%

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

      1. associate-/r* 66.0%

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

      2. sqrt-div 52.7%

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

      3. associate-*r/ 52.4%

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

   4. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

      2. associate-/l* 50.1%

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

   6. Simplified 50.1%

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

   if -5.0000000000000003e177 < (*.f64 V l) < -4.99999999999999989e-121

   1. Initial program 89.8%

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

      1. associate-/r* 78.2%

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

      2. div-inv 78.1%

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

      3. div-inv 78.2%

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

      4. associate-*l* 89.8%

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

   4. Applied egg-rr 89.8%

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

      1. frac-times 89.6%

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

      2. metadata-eval 89.6%

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

      3. div-inv 89.8%

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

      4. sqrt-undiv 0.0%

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

      5. clear-num 0.0%

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

      6. un-div-inv 0.0%

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

      7. sqrt-undiv 91.2%

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

   6. Applied egg-rr 91.2%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.6%

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

      1. sqrt-div 99.0%

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

      2. associate-*r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+305}\right):\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-253)
   (/ c0 (sqrt (/ (* V l) A)))
   (if (<= (* V l) 4e-319)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= (* V l) 1e+305)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (/ (sqrt (/ A l)) (sqrt V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-253) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 4e-319) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * (sqrt((A / l)) / sqrt(V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-253)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 4d-319) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 1d+305) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * (sqrt((a / l)) / sqrt(v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-253) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 4e-319) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * (Math.sqrt((A / l)) / Math.sqrt(V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e-253:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 4e-319:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * (math.sqrt((A / l)) / math.sqrt(V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-253)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / l)) / sqrt(V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-253)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 4e-319)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * (sqrt((A / l)) / sqrt(V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-253], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.0000000000000001e-253

   1. Initial program 74.1%

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

      1. associate-/r* 71.0%

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

      2. div-inv 71.0%

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

      3. div-inv 71.1%

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

      4. associate-*l* 74.3%

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

   4. Applied egg-rr 74.3%

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

      1. frac-times 74.0%

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

      2. metadata-eval 74.0%

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

      3. div-inv 74.1%

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

      4. sqrt-undiv 0.0%

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

      5. clear-num 0.0%

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

      6. un-div-inv 0.0%

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

      7. sqrt-undiv 75.7%

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

   6. Applied egg-rr 75.7%

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

   if -2.0000000000000001e-253 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/l/ 70.8%

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

   3. Simplified 70.8%

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

   if 4.0000049e-319 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.9%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 15.9%

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

      1. associate-/r* 68.5%

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

      2. div-inv 68.5%

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

      3. div-inv 68.5%

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

      4. associate-*l* 27.5%

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

   4. Applied egg-rr 27.5%

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

      1. *-commutative 27.5%

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

      2. associate-*r* 68.5%

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

      3. div-inv 68.5%

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

      4. div-inv 68.5%

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

      5. sqrt-div 75.5%

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

      6. associate-*r/ 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/l* 75.5%

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

   8. Simplified 75.5%

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

Alternative 9: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-253)
   (/ c0 (sqrt (/ (* V l) A)))
   (if (<= (* V l) 4e-319)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= (* V l) 1e+298)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-253) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 4e-319) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+298) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-253)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 4d-319) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 1d+298) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-253) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 4e-319) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 1e+298) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e-253:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 4e-319:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 1e+298:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-253)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 1e+298)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-253)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 4e-319)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 1e+298)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-253], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+298], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.0000000000000001e-253

   1. Initial program 74.1%

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

      1. associate-/r* 71.0%

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

      2. div-inv 71.0%

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

      3. div-inv 71.1%

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

      4. associate-*l* 74.3%

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

   4. Applied egg-rr 74.3%

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

      1. frac-times 74.0%

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

      2. metadata-eval 74.0%

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

      3. div-inv 74.1%

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

      4. sqrt-undiv 0.0%

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

      5. clear-num 0.0%

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

      6. un-div-inv 0.0%

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

      7. sqrt-undiv 75.7%

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

   6. Applied egg-rr 75.7%

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

   if -2.0000000000000001e-253 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/l/ 70.8%

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

   3. Simplified 70.8%

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

   if 4.0000049e-319 < (*.f64 V l) < 9.9999999999999996e297

   1. Initial program 87.8%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   if 9.9999999999999996e297 < (*.f64 V l)

   1. Initial program 25.2%

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

   3. Taylor expanded in c0 around 0 25.2%

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

      1. *-commutative 25.2%

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

      2. associate-/r* 72.0%

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

   5. Simplified 72.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+298}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{V}{A}}\\ \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{t\_0 \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot {\ell}^{-0.5}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ V A))))
   (if (<= (* V l) 0.0)
     (/ c0 (* t_0 (sqrt l)))
     (if (<= (* V l) 1e+305)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ (* c0 (pow l -0.5)) t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((V / A));
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (t_0 * sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0 * pow(l, -0.5)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((v / a))
    if ((v * l) <= 0.0d0) then
        tmp = c0 / (t_0 * sqrt(l))
    else if ((v * l) <= 1d+305) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = (c0 * (l ** (-0.5d0))) / t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((V / A));
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (t_0 * Math.sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0 * Math.pow(l, -0.5)) / t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((V / A))
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 / (t_0 * math.sqrt(l))
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0 * math.pow(l, -0.5)) / t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(V / A))
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(t_0 * sqrt(l)));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0 * (l ^ -0.5)) / t_0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((V / A));
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 / (t_0 * sqrt(l));
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0 * (l ^ -0.5)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(t$95$0 * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -0.0

   1. Initial program 66.5%

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

      1. associate-/r* 70.8%

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

      2. div-inv 70.8%

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

      3. div-inv 70.8%

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

      4. associate-*l* 66.4%

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

   4. Applied egg-rr 66.4%

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

      1. frac-times 66.2%

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

      2. metadata-eval 66.2%

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

      3. div-inv 66.5%

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

      4. sqrt-undiv 2.5%

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

      5. clear-num 2.5%

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

      6. un-div-inv 2.5%

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

      7. sqrt-undiv 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/l* 71.7%

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

   8. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. sqrt-prod 49.0%

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

   10. Applied egg-rr 49.0%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.6%

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

      1. sqrt-div 99.0%

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

      2. associate-*r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 15.9%

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

      1. associate-/r* 68.5%

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

      2. div-inv 68.5%

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

      3. div-inv 68.5%

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

      4. associate-*l* 27.5%

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

   4. Applied egg-rr 27.5%

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

      1. frac-times 15.9%

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

      2. metadata-eval 15.9%

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

      3. div-inv 15.9%

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

      4. sqrt-undiv 15.9%

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

      5. clear-num 15.9%

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

      6. un-div-inv 15.9%

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

      7. sqrt-undiv 15.9%

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

   6. Applied egg-rr 15.9%

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

      1. *-commutative 15.9%

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

      2. associate-/l* 64.0%

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

   8. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. sqrt-prod 75.3%

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

      3. times-frac 74.9%

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

      4. add-sqr-sqrt 74.9%

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

      5. sqrt-unprod 27.0%

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

      6. sqr-neg 27.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. pow1/2 0.0%

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

      10. pow-flip 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 27.0%

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

      13. sqr-neg 27.0%

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

      14. sqrt-unprod 75.1%

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

      15. add-sqr-sqrt 75.3%

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

      16. metadata-eval 75.3%

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

   10. Applied egg-rr 75.3%

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

       1. associate-*r/ 75.5%

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

       2. *-commutative 75.5%

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

   12. Simplified 75.5%

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

Alternative 11: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V} \cdot \sqrt{\frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) 1e+305)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (/ c0 (* (sqrt V) (sqrt (/ l A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / (sqrt(V) * sqrt((l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= 1d+305) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / (sqrt(v) * sqrt((l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / (Math.sqrt(V) * Math.sqrt((l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / (math.sqrt(V) * math.sqrt((l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / Float64(sqrt(V) * sqrt(Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / (sqrt(V) * sqrt((l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[V], $MachinePrecision] * N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -0.0

   1. Initial program 66.5%

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

      1. associate-/r* 70.8%

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

      2. div-inv 70.8%

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

      3. div-inv 70.8%

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

      4. associate-*l* 66.4%

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

   4. Applied egg-rr 66.4%

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

      1. frac-times 66.2%

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

      2. metadata-eval 66.2%

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

      3. div-inv 66.5%

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

      4. sqrt-undiv 2.5%

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

      5. clear-num 2.5%

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

      6. un-div-inv 2.5%

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

      7. sqrt-undiv 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/l* 71.7%

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

   8. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. sqrt-prod 49.0%

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

   10. Applied egg-rr 49.0%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.6%

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

      1. sqrt-div 99.0%

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

      2. associate-*r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 15.9%

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

      1. associate-/r* 68.5%

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

      2. div-inv 68.5%

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

      3. div-inv 68.5%

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

      4. associate-*l* 27.5%

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

   4. Applied egg-rr 27.5%

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

      1. frac-times 15.9%

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

      2. metadata-eval 15.9%

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

      3. div-inv 15.9%

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

      4. sqrt-undiv 15.9%

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

      5. clear-num 15.9%

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

      6. un-div-inv 15.9%

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

      7. sqrt-undiv 15.9%

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

   6. Applied egg-rr 15.9%

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

      1. associate-*r/ 63.8%

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

      2. *-commutative 63.8%

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

      3. sqrt-prod 75.3%

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

   8. Applied egg-rr 75.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V} \cdot \sqrt{\frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) 1e+305)
     (* c0 (/ (sqrt A) (sqrt (* V l))))
     (* (sqrt (/ A V)) (/ c0 (sqrt l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= 1d+305) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -0.0

   1. Initial program 66.5%

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

      1. associate-/r* 70.8%

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

      2. div-inv 70.8%

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

      3. div-inv 70.8%

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

      4. associate-*l* 66.4%

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

   4. Applied egg-rr 66.4%

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

      1. frac-times 66.2%

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

      2. metadata-eval 66.2%

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

      3. div-inv 66.5%

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

      4. sqrt-undiv 2.5%

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

      5. clear-num 2.5%

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

      6. un-div-inv 2.5%

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

      7. sqrt-undiv 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/l* 71.7%

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

   8. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. sqrt-prod 49.0%

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

   10. Applied egg-rr 49.0%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 87.6%

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

      1. sqrt-div 99.0%

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

      2. associate-*r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 15.9%

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

      1. associate-/r* 68.5%

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

      2. sqrt-div 75.3%

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

      3. associate-*r/ 74.7%

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

   4. Applied egg-rr 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-/l* 75.2%

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

   6. Simplified 75.2%

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

Alternative 13: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

Reproduce

herbie shell --seed 2024156 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))