Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 89.8%

Time: 12.2s

Alternatives: 10

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: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -1e-313)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 1e-312)
       (* c0 (/ 1.0 (sqrt (* V (/ l A)))))
       (* c0 (/ (sqrt A) (sqrt (* V l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -1e-313) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -1e-313) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / Math.sqrt((V * (l / A))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -1e-313:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-312:
		tmp = c0 * (1.0 / math.sqrt((V * (l / A))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -1e-313)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-312)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V * Float64(l / A)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -1e-313)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-312)
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-313], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-312], N[(c0 * N[(1.0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 41.0%

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

      1. frac-2neg 41.0%

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

      2. div-inv 41.0%

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

      3. distribute-rgt-neg-in 41.0%

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

   4. Applied egg-rr 41.0%

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

      1. un-div-inv 41.0%

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

      2. distribute-rgt-neg-out 41.0%

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

      3. frac-2neg 41.0%

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

      4. sqrt-div 0.0%

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

      5. clear-num 0.0%

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

      6. un-div-inv 0.0%

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

      7. sqrt-undiv 41.0%

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

      8. *-commutative 41.0%

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

      9. associate-*r/ 66.7%

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

      10. sqrt-prod 45.6%

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

      11. associate-/r* 45.6%

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

   6. Applied egg-rr 45.6%

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

      1. associate-/l/ 45.6%

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

   8. Simplified 45.6%

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

   if -inf.0 < (*.f64 V l) < -1.00000000001e-313

   1. Initial program 79.3%

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

      1. frac-2neg 79.3%

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

      2. sqrt-div 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. distribute-rgt-neg-out 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   6. Simplified 99.3%

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

   if -1.00000000001e-313 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 41.0%

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

      1. associate-/r* 68.8%

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

      2. clear-num 68.7%

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

      3. sqrt-div 68.9%

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

      4. metadata-eval 68.9%

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

      5. div-inv 68.9%

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

      6. clear-num 68.8%

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

   4. Applied egg-rr 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*l/ 41.0%

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

      3. associate-/l* 68.8%

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

   6. Simplified 68.8%

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

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 82.3%

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

      1. sqrt-div 93.1%

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

      2. div-inv 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 0.3× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+220) t_0 (* c0 (/ 1.0 (sqrt (* V (/ l A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+220) {
		tmp = t_0;
	} else {
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+220) then
        tmp = t_0
    else
        tmp = c0 * (1.0d0 / sqrt((v * (l / a))))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+220) {
		tmp = t_0;
	} else {
		tmp = c0 * (1.0 / Math.sqrt((V * (l / A))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+220:
		tmp = t_0
	else:
		tmp = c0 * (1.0 / math.sqrt((V * (l / A))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+220)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V * Float64(l / A)))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+220)
		tmp = t_0;
	else
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+220], t$95$0, N[(c0 * N[(1.0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.6%

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

      1. associate-/r* 67.0%

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

   3. Simplified 67.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4e220

   1. Initial program 98.6%

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

   if 4e220 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 49.7%

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

      1. associate-/r* 61.8%

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

      2. clear-num 61.7%

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

      3. sqrt-div 63.1%

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

      4. metadata-eval 63.1%

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

      5. div-inv 63.0%

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

      6. clear-num 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. *-commutative 65.2%

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

      2. associate-*l/ 51.0%

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

      3. associate-/l* 65.4%

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

   6. Simplified 65.4%

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

4. Final simplification 74.0%

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

Alternative 3: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+294)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+294)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+294))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+294)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+294):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+294))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+294)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+294]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.9999999999999999e294 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.5%

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

      1. associate-/r* 65.4%

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

   3. Simplified 65.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999999e294

   1. Initial program 98.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+264}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+264) t_0 (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+264) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+264) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+264) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+264:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+264)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+264)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+264], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.6%

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

      1. associate-/r* 67.0%

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

   3. Simplified 67.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000004e264

   1. Initial program 98.7%

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

   if 1.00000000000000004e264 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 45.7%

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

      1. frac-2neg 45.7%

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

      2. div-inv 45.7%

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

      3. distribute-rgt-neg-in 45.7%

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

   4. Applied egg-rr 45.7%

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

   5. Taylor expanded in c0 around 0 45.7%

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

      1. associate-/l/ 61.2%

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

   7. Simplified 61.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+264}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.6%

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

      1. associate-/r* 67.0%

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

   3. Simplified 67.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4e220

   1. Initial program 98.6%

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

   if 4e220 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 49.7%

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

      1. frac-2neg 49.7%

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

      2. div-inv 49.7%

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

      3. distribute-rgt-neg-in 49.7%

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

   4. Applied egg-rr 49.7%

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

      1. un-div-inv 49.7%

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

      2. distribute-rgt-neg-out 49.7%

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

      3. frac-2neg 49.7%

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

      4. sqrt-div 28.2%

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

      5. clear-num 28.2%

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

      6. un-div-inv 28.2%

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

      7. sqrt-undiv 51.0%

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

      8. *-commutative 51.0%

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

      9. associate-*r/ 65.1%

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

      10. clear-num 65.1%

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

      11. associate-*r/ 50.9%

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

      12. *-commutative 50.9%

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

      13. associate-/l* 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. associate-/r/ 65.4%

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

      2. metadata-eval 65.4%

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

      3. distribute-neg-frac 65.4%

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

      4. distribute-lft-neg-out 65.4%

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

      5. /-rgt-identity 65.4%

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

      6. times-frac 65.3%

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

      7. *-commutative 65.3%

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

      8. *-rgt-identity 65.3%

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

      9. *-commutative 65.3%

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

      10. neg-mul-1 65.3%

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

      11. distribute-frac-neg 65.3%

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

      12. distribute-frac-neg2 65.3%

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

      13. distribute-frac-neg2 65.3%

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

      14. remove-double-neg 65.3%

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

   8. Simplified 65.3%

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

4. Final simplification 74.0%

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

Alternative 6: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+143)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -2.4e-158)
     (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
     (if (<= (* V l) 1e-312)
       (* c0 (/ 1.0 (sqrt (* l (/ V A)))))
       (* c0 (/ (sqrt A) (sqrt (* V l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+143) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -2.4e-158) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+143)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-2.4d-158)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 1d-312) then
        tmp = c0 * (1.0d0 / sqrt((l * (v / a))))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+143) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -2.4e-158) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+143:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -2.4e-158:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 1e-312:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+143)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -2.4e-158)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 1e-312)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(l * Float64(V / A)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+143)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -2.4e-158)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 1e-312)
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+143], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2.4e-158], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-312], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e143

   1. Initial program 57.2%

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

      1. associate-/r* 66.6%

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

      2. sqrt-div 43.5%

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

      3. associate-*r/ 43.6%

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

   4. Applied egg-rr 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/l* 43.5%

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

   6. Simplified 43.5%

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

   if -1e143 < (*.f64 V l) < -2.40000000000000007e-158

   1. Initial program 93.6%

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

      1. clear-num 93.5%

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

      2. associate-/r/ 93.5%

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

      3. associate-/r* 93.6%

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

   4. Applied egg-rr 93.6%

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

   if -2.40000000000000007e-158 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 40.3%

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

      1. associate-/r* 57.9%

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

      2. clear-num 57.9%

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

      3. sqrt-div 60.5%

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

      4. metadata-eval 60.5%

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

      5. div-inv 60.4%

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

      6. clear-num 60.4%

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

   4. Applied egg-rr 60.4%

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

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 82.3%

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

      1. sqrt-div 93.1%

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

      2. div-inv 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+143}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+143)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -2.4e-158)
     (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
     (if (<= (* V l) 1e-312)
       (* c0 (/ 1.0 (sqrt (* l (/ V A)))))
       (* c0 (/ (sqrt A) (sqrt (* V l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+143) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2.4e-158) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+143)) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= (-2.4d-158)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 1d-312) then
        tmp = c0 * (1.0d0 / sqrt((l * (v / a))))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+143) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2.4e-158) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+143:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2.4e-158:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 1e-312:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+143)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2.4e-158)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 1e-312)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(l * Float64(V / A)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+143)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2.4e-158)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 1e-312)
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+143], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2.4e-158], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-312], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e143

   1. Initial program 57.2%

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

      1. frac-2neg 57.2%

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

      2. div-inv 57.3%

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

      3. distribute-rgt-neg-in 57.3%

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

   4. Applied egg-rr 57.3%

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

      1. un-div-inv 57.2%

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

      2. distribute-rgt-neg-out 57.2%

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

      3. frac-2neg 57.2%

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

      4. sqrt-div 0.0%

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

      5. clear-num 0.0%

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

      6. un-div-inv 0.0%

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

      7. sqrt-undiv 57.3%

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

      8. *-commutative 57.3%

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

      9. associate-*r/ 66.7%

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

      10. sqrt-prod 43.6%

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

      11. associate-/r* 43.5%

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

   6. Applied egg-rr 43.5%

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

      1. associate-/l/ 43.6%

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

   8. Simplified 43.6%

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

   if -1e143 < (*.f64 V l) < -2.40000000000000007e-158

   1. Initial program 93.6%

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

      1. clear-num 93.5%

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

      2. associate-/r/ 93.5%

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

      3. associate-/r* 93.6%

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

   4. Applied egg-rr 93.6%

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

   if -2.40000000000000007e-158 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 40.3%

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

      1. associate-/r* 57.9%

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

      2. clear-num 57.9%

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

      3. sqrt-div 60.5%

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

      4. metadata-eval 60.5%

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

      5. div-inv 60.4%

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

      6. clear-num 60.4%

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

   4. Applied egg-rr 60.4%

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

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 82.3%

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

      1. sqrt-div 93.1%

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

      2. div-inv 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+143}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2.4e-158)
   (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
   (if (<= (* V l) 1e-312)
     (* c0 (/ 1.0 (sqrt (* l (/ V A)))))
     (* c0 (/ (sqrt A) (sqrt (* V l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2.4e-158) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2.4d-158)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 1d-312) then
        tmp = c0 * (1.0d0 / sqrt((l * (v / a))))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2.4e-158) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 1e-312) {
		tmp = c0 * (1.0 / Math.sqrt((l * (V / A))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2.4e-158:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 1e-312:
		tmp = c0 * (1.0 / math.sqrt((l * (V / A))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2.4e-158)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 1e-312)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(l * Float64(V / A)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2.4e-158)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 1e-312)
		tmp = c0 * (1.0 / sqrt((l * (V / A))));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2.4e-158], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-312], N[(c0 * N[(1.0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -2.40000000000000007e-158

   1. Initial program 79.2%

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

      1. clear-num 79.2%

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

      2. associate-/r/ 79.2%

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

      3. associate-/r* 80.0%

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

   4. Applied egg-rr 80.0%

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

   if -2.40000000000000007e-158 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 40.3%

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

      1. associate-/r* 57.9%

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

      2. clear-num 57.9%

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

      3. sqrt-div 60.5%

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

      4. metadata-eval 60.5%

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

      5. div-inv 60.4%

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

      6. clear-num 60.4%

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

   4. Applied egg-rr 60.4%

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

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 82.3%

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

      1. sqrt-div 93.1%

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

      2. div-inv 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

3. Final simplification 73.2%

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

Alternative 10: 13.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ 0 \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 0.0)

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	return 0.0;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = 0.0d0
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	return 0.0;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	return 0.0

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return 0.0
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = 0.0;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := 0.0

Derivation

1. Initial program 73.2%

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

   1. *-commutative 73.2%

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

   2. add-log-exp 23.3%

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

   3. log-pow 22.9%

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

4. Applied egg-rr 22.9%

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

5. Taylor expanded in c0 around 0 12.1%

   \[\leadsto \log \color{blue}{1} \]

6. Final simplification 12.1%

   \[\leadsto 0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))