Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.4% → 87.7%

Time: 13.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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: 87.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. clear-num 76.3%

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

      2. sqrt-div 77.1%

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

      3. metadata-eval 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-/l* 77.1%

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

   3. Applied egg-rr 77.1%

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

      1. associate-/r/ 79.5%

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

   5. Simplified 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 79.2%

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

      3. un-div-inv 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. frac-2neg 79.2%

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

      2. sqrt-div 88.4%

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

      3. distribute-neg-frac 88.4%

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

   9. Applied egg-rr 88.4%

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

   if -4.999999999999985e-310 < V

   1. Initial program 67.4%

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

      1. pow1/2 67.4%

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

      2. clear-num 67.3%

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

      3. inv-pow 67.3%

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

      4. pow-pow 67.4%

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

      5. *-commutative 67.4%

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

      6. associate-/l* 66.2%

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

      7. metadata-eval 66.2%

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

   3. Applied egg-rr 66.2%

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

      1. associate-/r/ 65.5%

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

   5. Simplified 65.5%

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

      1. metadata-eval 65.5%

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

      2. pow-flip 65.4%

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

      3. pow1/2 65.4%

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

      4. associate-*l/ 67.2%

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

      5. *-commutative 67.2%

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

      6. sqrt-undiv 43.1%

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

      7. associate-/l* 43.1%

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

      8. sqrt-prod 53.6%

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

      9. times-frac 53.6%

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

   7. Applied egg-rr 53.6%

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

      1. associate-*l/ 53.7%

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

      2. *-lft-identity 53.7%

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

   9. Simplified 53.7%

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

4. Final simplification 71.3%

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

Alternative 2: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-221}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (* (sqrt l) (sqrt (/ V A))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-221)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 5e+299)
           (* c0 (* (sqrt A) (pow (* V l) -0.5)))
           (/ (* c0 (sqrt (/ A l))) (sqrt V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-221) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = (c0 * sqrt((A / l))) / sqrt(V);
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-221) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = (c0 * Math.sqrt((A / l))) / Math.sqrt(V);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-221:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 5e+299:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = (c0 * math.sqrt((A / l))) / math.sqrt(V)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-221)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 5e+299)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(Float64(c0 * sqrt(Float64(A / l))) / sqrt(V));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-221)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 5e+299)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = (c0 * sqrt((A / l))) / sqrt(V);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-221], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 5e+299], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[N[(A / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.99999999999999996e-221 < (*.f64 V l) < -0.0

   1. Initial program 41.7%

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

      1. pow1/2 41.7%

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

      2. clear-num 41.7%

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

      3. inv-pow 41.7%

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

      4. pow-pow 41.7%

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

      5. *-commutative 41.7%

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

      6. associate-/l* 60.4%

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

      7. metadata-eval 60.4%

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

   3. Applied egg-rr 60.4%

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

      1. associate-/r/ 60.3%

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

   5. Simplified 60.3%

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

      1. metadata-eval 60.3%

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

      2. pow-flip 60.3%

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

      3. pow1/2 60.3%

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

      4. un-div-inv 60.4%

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

      5. associate-/r/ 60.4%

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

      6. clear-num 59.3%

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

      7. associate-/r/ 60.4%

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

      8. clear-num 60.4%

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

      9. *-commutative 60.4%

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

   7. Applied egg-rr 60.4%

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

      1. *-commutative 60.4%

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

      2. sqrt-prod 46.2%

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

   9. Applied egg-rr 46.2%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999996e-221

   1. Initial program 87.8%

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

      1. frac-2neg 87.8%

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

      2. sqrt-div 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -0.0 < (*.f64 V l) < 5.0000000000000003e299

   1. Initial program 87.1%

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

      1. pow1/2 87.1%

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

      2. div-inv 86.6%

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

      3. unpow-prod-down 98.7%

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

      4. pow1/2 98.7%

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

      5. inv-pow 98.7%

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

      6. pow-pow 99.1%

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

      7. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   if 5.0000000000000003e299 < (*.f64 V l)

   1. Initial program 28.9%

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

      1. pow1/2 28.9%

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

      2. clear-num 28.9%

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

      3. inv-pow 28.9%

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

      4. pow-pow 28.9%

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

      5. *-commutative 28.9%

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

      6. associate-/l* 52.1%

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

      7. metadata-eval 52.1%

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

   3. Applied egg-rr 52.1%

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

      1. associate-/r/ 52.2%

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

   5. Simplified 52.2%

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

      1. metadata-eval 52.2%

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

      2. pow-flip 52.2%

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

      3. pow1/2 52.2%

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

      4. un-div-inv 52.4%

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

      5. associate-*l/ 28.9%

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

      6. *-commutative 28.9%

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

      7. sqrt-undiv 28.9%

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

      8. div-inv 28.9%

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

      9. associate-/r* 28.9%

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

      10. *-commutative 28.9%

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

      11. pow1/2 28.9%

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

      12. pow-flip 28.9%

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

      13. metadata-eval 28.9%

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

   7. Applied egg-rr 28.9%

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

      1. associate-/r* 28.9%

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

   9. Simplified 28.9%

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

       1. sqrt-prod 64.7%

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

       2. associate-*l* 64.7%

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

       3. add-sqr-sqrt 64.5%

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

       4. sqrt-unprod 64.7%

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

       5. pow-prod-up 64.7%

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

       6. metadata-eval 64.7%

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

       7. inv-pow 64.7%

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

       8. sqrt-prod 55.5%

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

       9. div-inv 55.6%

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

       10. sqrt-prod 52.4%

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

       11. *-commutative 52.4%

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

       12. associate-/r/ 52.4%

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

       13. un-div-inv 52.2%

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

       14. sqrt-div 55.6%

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

       15. associate-/r/ 55.7%

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

       16. sqrt-div 69.4%

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

       17. times-frac 69.4%

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

       18. *-un-lft-identity 69.4%

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

       19. sqrt-prod 28.9%

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

   11. Applied egg-rr 55.6%

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

       1. *-commutative 55.6%

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

       2. associate-*l/ 55.7%

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

   13. Simplified 55.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-221}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 5e+299)
     (* c0 (* (sqrt A) (pow (* V l) -0.5)))
     (/ (* c0 (sqrt (/ A l))) (sqrt V)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = (c0 * sqrt((A / l))) / sqrt(V);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d+299) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = (c0 * sqrt((a / l))) / sqrt(v)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = (c0 * Math.sqrt((A / l))) / Math.sqrt(V);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+299:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = (c0 * math.sqrt((A / l))) / math.sqrt(V)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+299)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(Float64(c0 * sqrt(Float64(A / l))) / sqrt(V));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+299)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = (c0 * sqrt((A / l))) / sqrt(V);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+299], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[N[(A / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.8%

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

      1. associate-/r* 70.0%

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

      2. sqrt-div 44.3%

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

   3. Applied egg-rr 44.3%

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

   if -0.0 < (*.f64 V l) < 5.0000000000000003e299

   1. Initial program 87.1%

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

      1. pow1/2 87.1%

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

      2. div-inv 86.6%

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

      3. unpow-prod-down 98.7%

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

      4. pow1/2 98.7%

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

      5. inv-pow 98.7%

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

      6. pow-pow 99.1%

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

      7. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   if 5.0000000000000003e299 < (*.f64 V l)

   1. Initial program 28.9%

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

      1. pow1/2 28.9%

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

      2. clear-num 28.9%

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

      3. inv-pow 28.9%

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

      4. pow-pow 28.9%

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

      5. *-commutative 28.9%

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

      6. associate-/l* 52.1%

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

      7. metadata-eval 52.1%

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

   3. Applied egg-rr 52.1%

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

      1. associate-/r/ 52.2%

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

   5. Simplified 52.2%

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

      1. metadata-eval 52.2%

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

      2. pow-flip 52.2%

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

      3. pow1/2 52.2%

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

      4. un-div-inv 52.4%

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

      5. associate-*l/ 28.9%

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

      6. *-commutative 28.9%

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

      7. sqrt-undiv 28.9%

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

      8. div-inv 28.9%

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

      9. associate-/r* 28.9%

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

      10. *-commutative 28.9%

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

      11. pow1/2 28.9%

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

      12. pow-flip 28.9%

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

      13. metadata-eval 28.9%

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

   7. Applied egg-rr 28.9%

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

      1. associate-/r* 28.9%

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

   9. Simplified 28.9%

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

       1. sqrt-prod 64.7%

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

       2. associate-*l* 64.7%

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

       3. add-sqr-sqrt 64.5%

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

       4. sqrt-unprod 64.7%

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

       5. pow-prod-up 64.7%

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

       6. metadata-eval 64.7%

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

       7. inv-pow 64.7%

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

       8. sqrt-prod 55.5%

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

       9. div-inv 55.6%

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

       10. sqrt-prod 52.4%

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

       11. *-commutative 52.4%

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

       12. associate-/r/ 52.4%

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

       13. un-div-inv 52.2%

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

       14. sqrt-div 55.6%

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

       15. associate-/r/ 55.7%

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

       16. sqrt-div 69.4%

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

       17. times-frac 69.4%

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

       18. *-un-lft-identity 69.4%

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

       19. sqrt-prod 28.9%

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

   11. Applied egg-rr 55.6%

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

       1. *-commutative 55.6%

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

       2. associate-*l/ 55.7%

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

   13. Simplified 55.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (* (sqrt (/ A V)) (sqrt (/ 1.0 l))))
   (if (<= (* V l) 5e+299)
     (* c0 (* (sqrt A) (pow (* V l) -0.5)))
     (/ (* c0 (sqrt (/ A l))) (sqrt V)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) * sqrt((1.0 / l)));
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = (c0 * sqrt((A / l))) / sqrt(V);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) * sqrt((1.0d0 / l)))
    else if ((v * l) <= 5d+299) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = (c0 * sqrt((a / l))) / sqrt(v)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) * Math.sqrt((1.0 / l)));
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = (c0 * Math.sqrt((A / l))) / Math.sqrt(V);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) * math.sqrt((1.0 / l)))
	elif (V * l) <= 5e+299:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = (c0 * math.sqrt((A / l))) / math.sqrt(V)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) * sqrt(Float64(1.0 / l))));
	elseif (Float64(V * l) <= 5e+299)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(Float64(c0 * sqrt(Float64(A / l))) / sqrt(V));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) * sqrt((1.0 / l)));
	elseif ((V * l) <= 5e+299)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = (c0 * sqrt((A / l))) / sqrt(V);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+299], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[N[(A / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.8%

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

      1. pow1/2 66.8%

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

      2. clear-num 66.7%

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

      3. inv-pow 66.7%

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

      4. pow-pow 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-/l* 70.5%

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

      7. metadata-eval 70.5%

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

   3. Applied egg-rr 70.5%

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

      1. associate-/r/ 73.5%

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

   5. Simplified 73.5%

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

      1. metadata-eval 73.5%

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

      2. pow-flip 73.3%

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

      3. pow1/2 73.3%

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

      4. metadata-eval 73.3%

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

      5. sqrt-div 73.0%

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

      6. associate-/r/ 70.0%

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

      7. clear-num 70.0%

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

      8. div-inv 70.0%

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

      9. sqrt-prod 44.3%

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

   7. Applied egg-rr 44.3%

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

   if -0.0 < (*.f64 V l) < 5.0000000000000003e299

   1. Initial program 87.1%

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

      1. pow1/2 87.1%

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

      2. div-inv 86.6%

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

      3. unpow-prod-down 98.7%

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

      4. pow1/2 98.7%

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

      5. inv-pow 98.7%

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

      6. pow-pow 99.1%

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

      7. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   if 5.0000000000000003e299 < (*.f64 V l)

   1. Initial program 28.9%

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

      1. pow1/2 28.9%

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

      2. clear-num 28.9%

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

      3. inv-pow 28.9%

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

      4. pow-pow 28.9%

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

      5. *-commutative 28.9%

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

      6. associate-/l* 52.1%

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

      7. metadata-eval 52.1%

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

   3. Applied egg-rr 52.1%

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

      1. associate-/r/ 52.2%

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

   5. Simplified 52.2%

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

      1. metadata-eval 52.2%

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

      2. pow-flip 52.2%

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

      3. pow1/2 52.2%

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

      4. un-div-inv 52.4%

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

      5. associate-*l/ 28.9%

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

      6. *-commutative 28.9%

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

      7. sqrt-undiv 28.9%

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

      8. div-inv 28.9%

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

      9. associate-/r* 28.9%

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

      10. *-commutative 28.9%

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

      11. pow1/2 28.9%

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

      12. pow-flip 28.9%

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

      13. metadata-eval 28.9%

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

   7. Applied egg-rr 28.9%

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

      1. associate-/r* 28.9%

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

   9. Simplified 28.9%

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

       1. sqrt-prod 64.7%

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

       2. associate-*l* 64.7%

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

       3. add-sqr-sqrt 64.5%

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

       4. sqrt-unprod 64.7%

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

       5. pow-prod-up 64.7%

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

       6. metadata-eval 64.7%

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

       7. inv-pow 64.7%

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

       8. sqrt-prod 55.5%

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

       9. div-inv 55.6%

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

       10. sqrt-prod 52.4%

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

       11. *-commutative 52.4%

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

       12. associate-/r/ 52.4%

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

       13. un-div-inv 52.2%

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

       14. sqrt-div 55.6%

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

       15. associate-/r/ 55.7%

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

       16. sqrt-div 69.4%

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

       17. times-frac 69.4%

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

       18. *-un-lft-identity 69.4%

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

       19. sqrt-prod 28.9%

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

   11. Applied egg-rr 55.6%

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

       1. *-commutative 55.6%

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

       2. associate-*l/ 55.7%

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

   13. Simplified 55.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{\ell}}}{\sqrt{V}}\\ \end{array} \]

Alternative 5: 84.1% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+287)))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (((V * l) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+287))) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (((V * l) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if ((V * l) <= 0.0) or not ((V * l) <= 1e+287):
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+287))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (((V * l) <= 0.0) || ~(((V * l) <= 1e+287)))
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+287]], $MachinePrecision]], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < -0.0 or 1.0000000000000001e287 < (*.f64 V l)

   1. Initial program 61.8%

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

      1. associate-/r* 67.5%

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

      2. sqrt-div 45.8%

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

   3. Applied egg-rr 45.8%

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

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

   1. Initial program 87.8%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 90.6%

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

   3. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-/l* 97.1%

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

      3. associate-/r/ 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+287}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 6: 84.1% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+287)))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (((V * l) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+287))) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (((V * l) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if ((V * l) <= 0.0) or not ((V * l) <= 1e+287):
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+287))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (((V * l) <= 0.0) || ~(((V * l) <= 1e+287)))
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+287]], $MachinePrecision]], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < -0.0 or 1.0000000000000001e287 < (*.f64 V l)

   1. Initial program 61.8%

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

      1. associate-/r* 67.5%

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

      2. sqrt-div 45.8%

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

   3. Applied egg-rr 45.8%

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

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

   1. Initial program 87.8%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 90.6%

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

   3. Applied egg-rr 90.6%

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

      1. associate-/l* 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+287}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 7: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -5e-310)
   (* c0 (/ 1.0 (/ (sqrt (- V)) (sqrt (/ (- A) l)))))
   (if (<= V 2.8e-8)
     (/ c0 (/ (sqrt (* V l)) (sqrt A)))
     (* (pow l -0.5) (/ c0 (sqrt (/ V A)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * (1.0 / (sqrt(-V) / sqrt((-A / l))));
	} else if (V <= 2.8e-8) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = pow(l, -0.5) * (c0 / sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-5d-310)) then
        tmp = c0 * (1.0d0 / (sqrt(-v) / sqrt((-a / l))))
    else if (v <= 2.8d-8) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = (l ** (-0.5d0)) * (c0 / sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * (1.0 / (Math.sqrt(-V) / Math.sqrt((-A / l))));
	} else if (V <= 2.8e-8) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = Math.pow(l, -0.5) * (c0 / Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -5e-310:
		tmp = c0 * (1.0 / (math.sqrt(-V) / math.sqrt((-A / l))))
	elif V <= 2.8e-8:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = math.pow(l, -0.5) * (c0 / math.sqrt((V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(Float64(-A) / l)))));
	elseif (V <= 2.8e-8)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64((l ^ -0.5) * Float64(c0 / sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -5e-310)
		tmp = c0 * (1.0 / (sqrt(-V) / sqrt((-A / l))));
	elseif (V <= 2.8e-8)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = (l ^ -0.5) * (c0 / sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -5e-310], N[(c0 * N[(1.0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[V, 2.8e-8], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, -0.5], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. clear-num 76.3%

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

      2. sqrt-div 77.1%

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

      3. metadata-eval 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-/l* 77.1%

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

   3. Applied egg-rr 77.1%

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

      1. associate-/r/ 79.5%

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

   5. Simplified 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 79.2%

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

      3. un-div-inv 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. frac-2neg 79.2%

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

      2. sqrt-div 88.4%

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

      3. distribute-neg-frac 88.4%

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

   9. Applied egg-rr 88.4%

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

   if -4.999999999999985e-310 < V < 2.7999999999999999e-8

   1. Initial program 74.8%

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

      1. sqrt-div 48.5%

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

      2. associate-*r/ 45.2%

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

   3. Applied egg-rr 45.2%

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

      1. associate-/l* 48.5%

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

   5. Simplified 48.5%

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

   if 2.7999999999999999e-8 < V

   1. Initial program 60.6%

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

      1. pow1/2 60.6%

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

      2. clear-num 60.5%

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

      3. inv-pow 60.5%

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

      4. pow-pow 60.5%

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

      5. *-commutative 60.5%

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

      6. associate-/l* 63.5%

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

      7. metadata-eval 63.5%

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

   3. Applied egg-rr 63.5%

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

      1. associate-/r/ 60.9%

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

   5. Simplified 60.9%

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

      1. metadata-eval 60.9%

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

      2. pow-flip 60.8%

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

      3. pow1/2 60.8%

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

      4. un-div-inv 60.9%

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

      5. associate-/r/ 63.5%

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

      6. clear-num 63.6%

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

      7. associate-/r/ 63.6%

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

      8. clear-num 63.6%

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

      9. *-commutative 63.6%

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

   7. Applied egg-rr 63.6%

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

      1. *-un-lft-identity 63.6%

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

      2. sqrt-prod 44.2%

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

      3. times-frac 42.9%

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

      4. pow1/2 42.9%

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

      5. pow-flip 43.0%

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

      6. metadata-eval 43.0%

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

   9. Applied egg-rr 43.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{-0.5} \cdot \frac{c0}{\sqrt{\frac{V}{A}}}\\ \end{array} \]

Alternative 8: 81.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 70.7%

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

      1. pow1/2 70.7%

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

      2. clear-num 70.7%

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

      3. inv-pow 70.7%

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

      4. pow-pow 71.7%

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

      5. *-commutative 71.7%

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

      6. associate-/l* 70.9%

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

      7. metadata-eval 70.9%

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

   3. Applied egg-rr 70.9%

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

      1. associate-/r/ 73.1%

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

   5. Simplified 73.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 73.1%

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

      1. associate-/r* 72.2%

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

      2. sqrt-div 84.0%

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

   3. Applied egg-rr 84.0%

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

4. Final simplification 78.5%

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

Alternative 9: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ t_1 := V \cdot \frac{\ell}{A}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\sqrt{t_1}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {t_1}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))) (t_1 (* V (/ l A))))
   (if (<= t_0 5e-308)
     (/ c0 (sqrt t_1))
     (if (<= t_0 4e+273) (/ c0 (sqrt (/ (* V l) A))) (* c0 (pow t_1 -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double t_1 = V * (l / A);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = c0 / sqrt(t_1);
	} else if (t_0 <= 4e+273) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 * pow(t_1, -0.5);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (v * l)
    t_1 = v * (l / a)
    if (t_0 <= 5d-308) then
        tmp = c0 / sqrt(t_1)
    else if (t_0 <= 4d+273) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 * (t_1 ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double t_1 = V * (l / A);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = c0 / Math.sqrt(t_1);
	} else if (t_0 <= 4e+273) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 * Math.pow(t_1, -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	t_1 = V * (l / A)
	tmp = 0
	if t_0 <= 5e-308:
		tmp = c0 / math.sqrt(t_1)
	elif t_0 <= 4e+273:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 * math.pow(t_1, -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	t_1 = Float64(V * Float64(l / A))
	tmp = 0.0
	if (t_0 <= 5e-308)
		tmp = Float64(c0 / sqrt(t_1));
	elseif (t_0 <= 4e+273)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 * (t_1 ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	t_1 = V * (l / A);
	tmp = 0.0;
	if (t_0 <= 5e-308)
		tmp = c0 / sqrt(t_1);
	elseif (t_0 <= 4e+273)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 * (t_1 ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-308], N[(c0 / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+273], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[t$95$1, -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.99999999999999955e-308

   1. Initial program 29.0%

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

      1. clear-num 29.0%

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

      2. sqrt-div 28.9%

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

      3. metadata-eval 28.9%

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

      4. *-commutative 28.9%

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

      5. associate-/l* 45.0%

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

   3. Applied egg-rr 45.0%

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

      1. associate-/r/ 46.6%

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

   5. Simplified 46.6%

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

      1. *-commutative 46.6%

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

      2. clear-num 46.6%

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

      3. un-div-inv 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. un-div-inv 46.7%

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

      2. pow1/2 46.7%

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

      3. associate-/r/ 45.1%

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

      4. *-commutative 45.1%

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

      5. metadata-eval 45.1%

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

      6. pow-prod-up 44.9%

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

      7. expm1-log1p-u 41.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 41.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 31.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 41.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 45.1%

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

       3. *-commutative 45.1%

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

       4. associate-*l/ 28.9%

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

       5. associate-*r/ 46.7%

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

   11. Simplified 46.7%

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

   if 4.99999999999999955e-308 < (/.f64 A (*.f64 V l)) < 3.99999999999999978e273

   1. Initial program 98.5%

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

      1. clear-num 98.5%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-/l* 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. associate-/r/ 89.1%

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

   5. Simplified 89.1%

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

      1. *-commutative 89.1%

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

      2. clear-num 88.8%

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

      3. un-div-inv 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. un-div-inv 89.1%

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

      2. pow1/2 89.1%

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

      3. associate-/r/ 88.1%

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

      4. *-commutative 88.1%

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

      5. metadata-eval 88.1%

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

      6. pow-prod-up 87.9%

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

      7. expm1-log1p-u 60.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 60.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 27.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 27.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 27.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 27.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 27.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 60.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 88.1%

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

       3. *-commutative 88.1%

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

       4. associate-*l/ 98.6%

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

       5. associate-*r/ 89.2%

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

   11. Simplified 89.2%

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

   12. Taylor expanded in V around 0 98.6%

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

   if 3.99999999999999978e273 < (/.f64 A (*.f64 V l))

   1. Initial program 38.3%

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

      1. pow1/2 38.3%

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

      2. clear-num 38.3%

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

      3. inv-pow 38.3%

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

      4. pow-pow 40.9%

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

      5. *-commutative 40.9%

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

      6. associate-/l* 50.9%

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

      7. metadata-eval 50.9%

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

   3. Applied egg-rr 50.9%

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

      1. associate-/r/ 50.9%

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

   5. Simplified 50.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 10: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 4 \cdot 10^{+290}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 4e+290)))
     (/ c0 (sqrt (* V (/ l A))))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+290)) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+290))) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+290)) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+290):
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+290))
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+290)))
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+290]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.00000000000000025e290 < (/.f64 A (*.f64 V l))

   1. Initial program 30.3%

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

      1. clear-num 30.3%

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

      2. sqrt-div 31.5%

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

      3. metadata-eval 31.5%

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

      4. *-commutative 31.5%

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

      5. associate-/l* 46.4%

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

   3. Applied egg-rr 46.4%

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

      1. associate-/r/ 46.4%

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

   5. Simplified 46.4%

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

      1. *-commutative 46.4%

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

      2. clear-num 46.4%

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

      3. un-div-inv 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. un-div-inv 46.5%

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

      2. pow1/2 46.5%

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

      3. associate-/r/ 46.4%

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

      4. *-commutative 46.4%

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

      5. metadata-eval 46.4%

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

      6. pow-prod-up 46.3%

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

      7. expm1-log1p-u 37.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 37.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 30.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 30.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 30.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 30.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 30.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 30.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 37.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 46.4%

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

       3. *-commutative 46.4%

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

       4. associate-*l/ 31.5%

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

       5. associate-*r/ 46.5%

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

   11. Simplified 46.5%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000025e290

   1. Initial program 98.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+290}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 11: 79.7% accurate, 0.9× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 1e+270) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 1e+270) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 1d+270) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 1e+270) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 1e+270:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 1e+270)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 1e+270)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+270], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

      1. clear-num 25.2%

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

      2. sqrt-div 25.2%

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

      3. metadata-eval 25.2%

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

      4. *-commutative 25.2%

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

      5. associate-/l* 43.7%

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

   3. Applied egg-rr 43.7%

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

      1. associate-/r/ 43.8%

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

   5. Simplified 43.8%

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

      1. *-commutative 43.8%

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

      2. clear-num 43.8%

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

      3. un-div-inv 43.7%

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

   7. Applied egg-rr 43.7%

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

      1. un-div-inv 43.9%

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

      2. pow1/2 43.9%

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

      3. associate-/r/ 43.8%

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

      4. *-commutative 43.8%

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

      5. metadata-eval 43.8%

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

      6. pow-prod-up 43.7%

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

      7. expm1-log1p-u 39.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 39.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 29.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 29.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 40.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 43.8%

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

       3. *-commutative 43.8%

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

       4. associate-*l/ 25.2%

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

       5. associate-*r/ 43.9%

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

   11. Simplified 43.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1e270

   1. Initial program 98.6%

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

   if 1e270 < (/.f64 A (*.f64 V l))

   1. Initial program 39.6%

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

      1. pow1/2 39.6%

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

      2. clear-num 39.6%

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

      3. inv-pow 39.6%

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

      4. pow-pow 42.2%

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

      5. *-commutative 42.2%

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

      6. associate-/l* 52.0%

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

      7. metadata-eval 52.0%

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

   3. Applied egg-rr 52.0%

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

      1. associate-/r/ 49.9%

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

   5. Simplified 49.9%

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

      1. metadata-eval 49.9%

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

      2. pow-flip 49.8%

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

      3. pow1/2 49.8%

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

      4. un-div-inv 49.8%

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

      5. associate-/r/ 51.9%

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

      6. clear-num 48.0%

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

      7. associate-/r/ 51.9%

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

      8. clear-num 51.9%

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

      9. *-commutative 51.9%

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

   7. Applied egg-rr 51.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+270}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 12: 79.7% accurate, 0.9× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 1e+270) (* c0 (sqrt t_0)) (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 1e+270) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 1d+270) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 1e+270) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 1e+270:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 1e+270)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 1e+270)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+270], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

      1. clear-num 25.2%

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

      2. sqrt-div 25.2%

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

      3. metadata-eval 25.2%

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

      4. *-commutative 25.2%

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

      5. associate-/l* 43.7%

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

   3. Applied egg-rr 43.7%

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

      1. associate-/r/ 43.8%

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

   5. Simplified 43.8%

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

      1. *-commutative 43.8%

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

      2. clear-num 43.8%

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

      3. un-div-inv 43.7%

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

   7. Applied egg-rr 43.7%

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

      1. un-div-inv 43.9%

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

      2. pow1/2 43.9%

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

      3. associate-/r/ 43.8%

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

      4. *-commutative 43.8%

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

      5. metadata-eval 43.8%

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

      6. pow-prod-up 43.7%

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

      7. expm1-log1p-u 39.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 39.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 29.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 29.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 29.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 40.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 43.8%

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

       3. *-commutative 43.8%

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

       4. associate-*l/ 25.2%

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

       5. associate-*r/ 43.9%

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

   11. Simplified 43.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1e270

   1. Initial program 98.6%

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

   if 1e270 < (/.f64 A (*.f64 V l))

   1. Initial program 39.6%

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

      1. associate-/r* 48.0%

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

      2. sqrt-div 30.8%

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

   3. Applied egg-rr 30.8%

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

      1. frac-2neg 30.8%

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

      2. clear-num 30.8%

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

      3. frac-2neg 30.8%

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

      4. sqrt-undiv 51.9%

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

      5. un-div-inv 51.9%

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

      6. clear-num 51.9%

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

      7. div-inv 51.9%

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

      8. clear-num 51.9%

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

      9. un-div-inv 51.9%

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

   5. Applied egg-rr 51.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+270}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 13: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 10^{+270}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-308)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 1e+270)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 1e+270) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-308) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 1d+270) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((l / (a / v)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 1e+270) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-308:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 1e+270:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-308)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 1e+270)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-308)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 1e+270)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-308], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+270], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.99999999999999955e-308

   1. Initial program 29.0%

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

      1. clear-num 29.0%

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

      2. sqrt-div 28.9%

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

      3. metadata-eval 28.9%

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

      4. *-commutative 28.9%

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

      5. associate-/l* 45.0%

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

   3. Applied egg-rr 45.0%

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

      1. associate-/r/ 46.6%

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

   5. Simplified 46.6%

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

      1. *-commutative 46.6%

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

      2. clear-num 46.6%

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

      3. un-div-inv 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. un-div-inv 46.7%

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

      2. pow1/2 46.7%

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

      3. associate-/r/ 45.1%

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

      4. *-commutative 45.1%

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

      5. metadata-eval 45.1%

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

      6. pow-prod-up 44.9%

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

      7. expm1-log1p-u 41.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 41.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 31.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 41.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 45.1%

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

       3. *-commutative 45.1%

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

       4. associate-*l/ 28.9%

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

       5. associate-*r/ 46.7%

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

   11. Simplified 46.7%

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

   if 4.99999999999999955e-308 < (/.f64 A (*.f64 V l)) < 1e270

   1. Initial program 98.5%

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

      1. clear-num 98.5%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-/l* 88.0%

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

   3. Applied egg-rr 88.0%

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

      1. associate-/r/ 89.7%

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

   5. Simplified 89.7%

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

      1. *-commutative 89.7%

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

      2. clear-num 89.4%

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

      3. un-div-inv 89.5%

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

   7. Applied egg-rr 89.5%

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

      1. un-div-inv 89.6%

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

      2. pow1/2 89.6%

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

      3. associate-/r/ 88.0%

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

      4. *-commutative 88.0%

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

      5. metadata-eval 88.0%

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

      6. pow-prod-up 87.8%

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

      7. expm1-log1p-u 60.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)\right)} \]

      8. associate-/l/ 60.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)\right) \]

      9. expm1-udef 27.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}\right)} - 1} \]

      10. associate-/l/ 27.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{0.25} \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{0.25}}}\right)} - 1 \]

      11. pow-sqr 27.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{{\left(\ell \cdot \frac{V}{A}\right)}^{\left(2 \cdot 0.25\right)}}}\right)} - 1 \]

      12. metadata-eval 27.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{{\left(\ell \cdot \frac{V}{A}\right)}^{\color{blue}{0.5}}}\right)} - 1 \]

      13. pow1/2 27.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\ell \cdot \frac{V}{A}}}}\right)} - 1 \]

   9. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 60.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

       2. expm1-log1p 88.0%

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

       3. *-commutative 88.0%

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

       4. associate-*l/ 98.6%

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

       5. associate-*r/ 89.8%

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

   11. Simplified 89.8%

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

   12. Taylor expanded in V around 0 98.6%

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

   if 1e270 < (/.f64 A (*.f64 V l))

   1. Initial program 39.6%

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

      1. associate-/r* 48.0%

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

      2. sqrt-div 30.8%

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

   3. Applied egg-rr 30.8%

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

      1. frac-2neg 30.8%

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

      2. clear-num 30.8%

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

      3. frac-2neg 30.8%

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

      4. sqrt-undiv 51.9%

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

      5. un-div-inv 51.9%

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

      6. clear-num 51.9%

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

      7. div-inv 51.9%

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

      8. clear-num 51.9%

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

      9. un-div-inv 51.9%

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

   5. Applied egg-rr 51.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+270}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 14: 77.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.2%

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

      1. sqrt-div 26.3%

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

      2. associate-*r/ 26.3%

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

   3. Applied egg-rr 26.3%

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

      1. associate-/l* 26.3%

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

   5. Simplified 26.3%

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

      1. div-inv 26.3%

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

      2. sqrt-undiv 25.2%

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

      3. *-commutative 25.2%

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

      4. associate-*l/ 43.8%

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

      5. *-commutative 43.8%

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

      6. metadata-eval 43.8%

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

      7. sqrt-div 43.8%

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

      8. associate-/r/ 43.8%

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

      9. clear-num 43.8%

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

   7. Applied egg-rr 43.8%

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

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

   1. Initial program 85.0%

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

4. Final simplification 76.0%

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

Alternative 15: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

2. Final simplification 71.9%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))