Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 87.3%

Time: 11.3s

Alternatives: 12

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: 73.3% 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: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -2e-310)
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (* c0 (/ (/ (sqrt A) (sqrt V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2e-310) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else {
		tmp = c0 * ((sqrt(A) / sqrt(V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-2d-310)) then
        tmp = (sqrt((-a / l)) / sqrt(-v)) * c0
    else
        tmp = c0 * ((sqrt(a) / sqrt(v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2e-310) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else {
		tmp = c0 * ((Math.sqrt(A) / Math.sqrt(V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -2e-310:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	else:
		tmp = c0 * ((math.sqrt(A) / math.sqrt(V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	else
		tmp = Float64(c0 * Float64(Float64(sqrt(A) / sqrt(V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -2e-310)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	else
		tmp = c0 * ((sqrt(A) / sqrt(V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -1.999999999999994e-310

   1. Initial program 64.3%

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

   3. Taylor expanded in c0 around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-/r* 72.5%

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

   5. Simplified 72.5%

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

      1. frac-2neg 72.5%

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

      2. sqrt-div 88.5%

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

      3. distribute-neg-frac2 88.5%

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

   7. Applied egg-rr 88.5%

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

   if -1.999999999999994e-310 < V

   1. Initial program 74.6%

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

      1. associate-/r* 79.2%

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

      2. sqrt-div 47.0%

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

      3. associate-*r/ 45.6%

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

   4. Applied egg-rr 45.6%

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

      1. associate-/l* 47.0%

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

   6. Simplified 47.0%

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

      1. sqrt-div 50.8%

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

      2. div-inv 50.9%

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

   8. Applied egg-rr 50.9%

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

      1. associate-*r/ 50.8%

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

      2. *-rgt-identity 50.8%

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

   10. Simplified 50.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+180}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 1e+180)))
     (/ c0 (sqrt (* V (/ l A))))
     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 <= 1e+180)) {
		tmp = c0 / sqrt((V * (l / A)));
	} 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 <= 1d+180))) then
        tmp = c0 / sqrt((v * (l / a)))
    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 <= 1e+180)) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} 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 <= 1e+180):
		tmp = c0 / math.sqrt((V * (l / A)))
	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 <= 1e+180))
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	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 <= 1e+180)))
		tmp = c0 / sqrt((V * (l / A)));
	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, 1e+180]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $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 1e180 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 62.2%

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

      1. associate-/r* 71.7%

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

      2. clear-num 71.3%

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

      3. sqrt-div 72.4%

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

      4. metadata-eval 72.4%

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

      5. clear-num 71.2%

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

      6. associate-/r* 61.6%

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

      7. clear-num 62.1%

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

      8. associate-/l* 74.7%

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

   4. Applied egg-rr 74.7%

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

      1. un-div-inv 74.8%

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

      2. *-commutative 74.8%

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

      3. sqrt-prod 45.3%

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

      4. sqrt-div 26.7%

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

      5. associate-/r/ 26.7%

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

      6. sqrt-div 44.8%

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

      7. sqrt-undiv 72.4%

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

      8. div-inv 72.4%

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

      9. clear-num 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. associate-*r/ 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-/l* 74.8%

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

   8. Simplified 74.8%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e180

   1. Initial program 99.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+180}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+276}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -4e+220)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -5e-314)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (sqrt (* A (* (/ c0 l) (/ c0 V))))
       (if (<= (* V l) 5e+276)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 (sqrt (* V (/ l A)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e+220) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -5e-314) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	} else if ((V * l) <= 5e+276) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} 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) :: tmp
    if ((v * l) <= (-4d+220)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-5d-314)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = sqrt((a * ((c0 / l) * (c0 / v))))
    else if ((v * l) <= 5d+276) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    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 tmp;
	if ((V * l) <= -4e+220) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -5e-314) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	} else if ((V * l) <= 5e+276) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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):
	tmp = 0
	if (V * l) <= -4e+220:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -5e-314:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	elif (V * l) <= 5e+276:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	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)
	tmp = 0.0
	if (Float64(V * l) <= -4e+220)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -5e-314)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	elseif (Float64(V * l) <= 5e+276)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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)
	tmp = 0.0;
	if ((V * l) <= -4e+220)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -5e-314)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	elseif ((V * l) <= 5e+276)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], -4e+220], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-314], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+276], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4e220

   1. Initial program 48.9%

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

      1. associate-/r* 84.5%

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

      2. sqrt-div 38.7%

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

      3. associate-*r/ 38.7%

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

   4. Applied egg-rr 38.7%

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

      1. associate-/l* 38.7%

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

   6. Simplified 38.7%

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

   if -4e220 < (*.f64 V l) < -4.99999999982e-314

   1. Initial program 90.8%

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

      1. frac-2neg 90.8%

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

      2. sqrt-div 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-rgt-neg-in 99.2%

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

   4. Applied egg-rr 99.2%

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

   if -4.99999999982e-314 < (*.f64 V l) < 0.0

   1. Initial program 33.0%

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

      1. add-sqr-sqrt 15.1%

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

      2. sqrt-unprod 15.3%

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

      3. *-commutative 15.3%

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

      4. *-commutative 15.3%

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

      5. swap-sqr 14.8%

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

      6. add-sqr-sqrt 14.8%

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

      7. pow2 14.8%

         \[\leadsto \sqrt{\frac{A}{V \cdot \ell} \cdot \color{blue}{{c0}^{2}}} \]

   4. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/r* 17.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}} \cdot {c0}^{2}} \]

   6. Simplified 17.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell} \cdot {c0}^{2}}} \]

   7. Taylor expanded in A around 0 14.7%

      \[\leadsto \sqrt{\color{blue}{\frac{A \cdot {c0}^{2}}{V \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 14.8%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]

      2. *-commutative 14.8%

         \[\leadsto \sqrt{A \cdot \frac{{c0}^{2}}{\color{blue}{\ell \cdot V}}} \]

      3. associate-/r* 15.6%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{{c0}^{2}}{\ell}}{V}}} \]

   9. Simplified 15.6%

      \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{{c0}^{2}}{\ell}}{V}}} \]
   10. Step-by-step derivation

       1. associate-/r* 14.8%

          \[\leadsto \sqrt{A \cdot \color{blue}{\frac{{c0}^{2}}{\ell \cdot V}}} \]

       2. unpow2 14.8%

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

       3. times-frac 26.4%

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

   11. Applied egg-rr 26.4%

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

   if 0.0 < (*.f64 V l) < 5.00000000000000001e276

   1. Initial program 80.9%

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

      1. sqrt-div 99.6%

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

      2. associate-*r/ 97.2%

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

   4. Applied egg-rr 97.2%

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

      1. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

   if 5.00000000000000001e276 < (*.f64 V l)

   1. Initial program 54.8%

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

      1. associate-/r* 76.8%

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

      2. clear-num 76.8%

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

      3. sqrt-div 76.7%

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

      4. metadata-eval 76.7%

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

      5. clear-num 76.7%

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

      6. associate-/r* 54.7%

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

      7. clear-num 54.7%

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

      8. associate-/l* 76.8%

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

   4. Applied egg-rr 76.8%

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

      1. un-div-inv 76.9%

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

      2. *-commutative 76.9%

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

      3. sqrt-prod 58.0%

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

      4. sqrt-div 63.6%

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

      5. associate-/r/ 63.4%

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

      6. sqrt-div 61.7%

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

      7. sqrt-undiv 76.8%

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

      8. div-inv 76.8%

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

      9. clear-num 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. associate-*r/ 54.8%

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

      2. *-commutative 54.8%

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

      3. associate-/l* 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+220}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-314}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+276}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{-A}{V}}}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-312}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ (- A) V)) (sqrt (- l))))))
   (if (<= l -4.5e+74)
     t_0
     (if (<= l -5.3e-244)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (if (<= l -5e-312) t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((-A / V)) / sqrt(-l));
	double tmp;
	if (l <= -4.5e+74) {
		tmp = t_0;
	} else if (l <= -5.3e-244) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else if (l <= -5e-312) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((-a / v)) / sqrt(-l))
    if (l <= (-4.5d+74)) then
        tmp = t_0
    else if (l <= (-5.3d-244)) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else if (l <= (-5d-312)) then
        tmp = t_0
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((-A / V)) / Math.sqrt(-l));
	double tmp;
	if (l <= -4.5e+74) {
		tmp = t_0;
	} else if (l <= -5.3e-244) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else if (l <= -5e-312) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((-A / V)) / math.sqrt(-l))
	tmp = 0
	if l <= -4.5e+74:
		tmp = t_0
	elif l <= -5.3e-244:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	elif l <= -5e-312:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / V)) / sqrt(Float64(-l))))
	tmp = 0.0
	if (l <= -4.5e+74)
		tmp = t_0;
	elseif (l <= -5.3e-244)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	elseif (l <= -5e-312)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((-A / V)) / sqrt(-l));
	tmp = 0.0;
	if (l <= -4.5e+74)
		tmp = t_0;
	elseif (l <= -5.3e-244)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	elseif (l <= -5e-312)
		tmp = t_0;
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[((-A) / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+74], t$95$0, If[LessEqual[l, -5.3e-244], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-312], t$95$0, N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5e74 or -5.3000000000000001e-244 < l < -5.0000000000022e-312

   1. Initial program 62.7%

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

      1. associate-/r* 74.5%

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

      2. frac-2neg 74.5%

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

      3. sqrt-div 94.0%

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

      4. distribute-neg-frac2 94.0%

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

   4. Applied egg-rr 94.0%

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

   if -4.5e74 < l < -5.3000000000000001e-244

   1. Initial program 74.0%

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

      1. sqrt-div 41.3%

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

      2. associate-*r/ 39.9%

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

   4. Applied egg-rr 39.9%

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

      1. associate-/l* 41.3%

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

   6. Simplified 41.3%

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

   if -5.0000000000022e-312 < l

   1. Initial program 69.9%

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

      1. associate-/r* 75.1%

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

      2. sqrt-div 88.8%

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

      3. associate-*r/ 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+74}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{V}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5.3 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{V}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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 (or (<= (* V l) 0.0) (not (<= (* V l) 5e+276)))
   (/ c0 (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) <= 0.0) || !((V * l) <= 5e+276)) {
		tmp = c0 / sqrt((V * (l / 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) <= 0.0d0) .or. (.not. ((v * l) <= 5d+276))) then
        tmp = c0 / sqrt((v * (l / 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) <= 0.0) || !((V * l) <= 5e+276)) {
		tmp = c0 / 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) <= 0.0) or not ((V * l) <= 5e+276):
		tmp = c0 / 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) <= 0.0) || !(Float64(V * l) <= 5e+276))
		tmp = Float64(c0 / 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) <= 0.0) || ~(((V * l) <= 5e+276)))
		tmp = c0 / 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[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 5e+276]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(V * N[(l / A), $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 2 regimes

2. if (*.f64 V l) < 0.0 or 5.00000000000000001e276 < (*.f64 V l)

   1. Initial program 64.3%

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

      1. associate-/r* 76.6%

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

      2. clear-num 76.5%

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

      3. sqrt-div 77.3%

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

      4. metadata-eval 77.3%

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

      5. clear-num 76.4%

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

      6. associate-/r* 64.2%

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

      7. clear-num 64.3%

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

      8. associate-/l* 76.9%

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

   4. Applied egg-rr 76.9%

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

      1. un-div-inv 77.0%

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

      2. *-commutative 77.0%

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

      3. sqrt-prod 45.7%

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

      4. sqrt-div 14.2%

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

      5. associate-/r/ 14.2%

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

      6. sqrt-div 44.3%

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

      7. sqrt-undiv 77.4%

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

      8. div-inv 77.4%

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

      9. clear-num 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. associate-*r/ 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-/l* 77.0%

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

   8. Simplified 77.0%

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

   if 0.0 < (*.f64 V l) < 5.00000000000000001e276

   1. Initial program 80.9%

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

      1. sqrt-div 99.6%

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

      2. associate-*r/ 97.2%

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

   4. Applied egg-rr 97.2%

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

      1. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-239}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-239)
   (* c0 (/ (sqrt A) (sqrt (* V l))))
   (if (<= l 1.4e-289)
     (sqrt (* A (* (/ c0 l) (/ c0 V))))
     (* c0 (/ (sqrt (/ A V)) (sqrt l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-239) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else if (l <= 1.4e-289) {
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-239)) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else if (l <= 1.4d-289) then
        tmp = sqrt((a * ((c0 / l) * (c0 / v))))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-239) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else if (l <= 1.4e-289) {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-239:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	elif l <= 1.4e-289:
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-239)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	elseif (l <= 1.4e-289)
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-239)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	elseif (l <= 1.4e-289)
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-239], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.4e-289], N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e-239

   1. Initial program 73.7%

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

      1. sqrt-div 37.6%

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

      2. associate-*r/ 36.7%

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

   4. Applied egg-rr 36.7%

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

      1. associate-/l* 37.6%

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

   6. Simplified 37.6%

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

   if -5e-239 < l < 1.39999999999999993e-289

   1. Initial program 49.5%

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

      1. add-sqr-sqrt 35.0%

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

      2. sqrt-unprod 27.2%

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

      3. *-commutative 27.2%

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

      4. *-commutative 27.2%

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

      5. swap-sqr 22.7%

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

      6. add-sqr-sqrt 22.7%

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

      7. pow2 22.7%

         \[\leadsto \sqrt{\frac{A}{V \cdot \ell} \cdot \color{blue}{{c0}^{2}}} \]

   4. Applied egg-rr 22.7%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/r* 19.4%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}} \cdot {c0}^{2}} \]

   6. Simplified 19.4%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell} \cdot {c0}^{2}}} \]

   7. Taylor expanded in A around 0 19.2%

      \[\leadsto \sqrt{\color{blue}{\frac{A \cdot {c0}^{2}}{V \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 23.3%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]

      2. *-commutative 23.3%

         \[\leadsto \sqrt{A \cdot \frac{{c0}^{2}}{\color{blue}{\ell \cdot V}}} \]

      3. associate-/r* 23.8%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{{c0}^{2}}{\ell}}{V}}} \]

   9. Simplified 23.8%

      \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{{c0}^{2}}{\ell}}{V}}} \]
   10. Step-by-step derivation

       1. associate-/r* 23.3%

          \[\leadsto \sqrt{A \cdot \color{blue}{\frac{{c0}^{2}}{\ell \cdot V}}} \]

       2. unpow2 23.3%

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

       3. times-frac 32.1%

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

   11. Applied egg-rr 32.1%

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

   if 1.39999999999999993e-289 < l

   1. Initial program 69.7%

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

      1. associate-/r* 75.9%

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

      2. sqrt-div 89.3%

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

      3. associate-*r/ 85.8%

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

   4. Applied egg-rr 85.8%

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

      1. associate-/l* 89.3%

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

   6. Simplified 89.3%

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

Alternative 7: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -1.15 \cdot 10^{-293}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -1.15e-293)
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -1.15e-293) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-1.15d-293)) then
        tmp = (sqrt((-a / l)) / sqrt(-v)) * c0
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -1.15e-293) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -1.15e-293:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	else:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -1.15e-293)
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	else
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -1.15e-293)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	else
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -1.15e-293], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -1.14999999999999998e-293

   1. Initial program 64.8%

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

   3. Taylor expanded in c0 around 0 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-/r* 73.0%

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

   5. Simplified 73.0%

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

      1. frac-2neg 73.0%

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

      2. sqrt-div 88.5%

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

      3. distribute-neg-frac2 88.5%

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

   7. Applied egg-rr 88.5%

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

   if -1.14999999999999998e-293 < V

   1. Initial program 74.0%

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

      1. associate-/r* 78.7%

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

      2. clear-num 78.6%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. clear-num 78.5%

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

      6. associate-/r* 73.3%

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

      7. clear-num 74.0%

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

      8. associate-/l* 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. un-div-inv 77.9%

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

      2. *-commutative 77.9%

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

      3. sqrt-prod 87.1%

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

      4. sqrt-div 50.5%

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

      5. associate-/r/ 50.5%

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

      6. sqrt-div 46.7%

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

      7. sqrt-undiv 79.3%

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

      8. div-inv 79.3%

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

      9. clear-num 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-*r/ 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-/l* 77.9%

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

   8. Simplified 77.9%

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

      1. add-sqr-sqrt 33.5%

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

      2. associate-*r/ 33.5%

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

      3. *-commutative 33.5%

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

      4. associate-*r/ 37.1%

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

      5. sqrt-prod 19.3%

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

      6. times-frac 19.3%

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

   10. Applied egg-rr 19.3%

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

       1. associate-*l/ 18.6%

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

       2. associate-*r/ 18.6%

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

       3. rem-square-sqrt 45.2%

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

       4. associate-/r* 46.7%

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

   12. Simplified 46.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -1.15 \cdot 10^{-293}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+297)))
     (sqrt (* A (* (/ c0 l) (/ c0 V))))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && 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 <= 2e+297)) {
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	} else {
		tmp = c0 * sqrt(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 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+297))) then
        tmp = sqrt((a * ((c0 / l) * (c0 / v))))
    else
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+297)) {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+297):
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, 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 <= 2e+297))
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	else
		tmp = Float64(c0 * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+297)))
		tmp = sqrt((A * ((c0 / l) * (c0 / V))));
	else
		tmp = c0 * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+297]], $MachinePrecision]], N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $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 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 39.5%

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

      1. add-sqr-sqrt 30.0%

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

      2. sqrt-unprod 30.0%

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

      3. *-commutative 30.0%

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

      4. *-commutative 30.0%

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

      5. swap-sqr 29.4%

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

      6. add-sqr-sqrt 29.4%

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

      7. pow2 29.4%

         \[\leadsto \sqrt{\frac{A}{V \cdot \ell} \cdot \color{blue}{{c0}^{2}}} \]

   4. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/r* 31.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}} \cdot {c0}^{2}} \]

   6. Simplified 31.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell} \cdot {c0}^{2}}} \]

   7. Taylor expanded in A around 0 32.6%

      \[\leadsto \sqrt{\color{blue}{\frac{A \cdot {c0}^{2}}{V \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 32.6%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]

      2. *-commutative 32.6%

         \[\leadsto \sqrt{A \cdot \frac{{c0}^{2}}{\color{blue}{\ell \cdot V}}} \]

      3. associate-/r* 33.1%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{{c0}^{2}}{\ell}}{V}}} \]

   9. Simplified 33.1%

      \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{{c0}^{2}}{\ell}}{V}}} \]
   10. Step-by-step derivation

       1. associate-/r* 32.6%

          \[\leadsto \sqrt{A \cdot \color{blue}{\frac{{c0}^{2}}{\ell \cdot V}}} \]

       2. unpow2 32.6%

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

       3. times-frac 44.5%

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

   11. Applied egg-rr 44.5%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2e297

   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 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;\sqrt{A \cdot \left(c0 \cdot \frac{\frac{c0}{\ell}}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+297)))
     (sqrt (* A (* c0 (/ (/ c0 l) V))))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && 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 <= 2e+297)) {
		tmp = sqrt((A * (c0 * ((c0 / l) / V))));
	} else {
		tmp = c0 * sqrt(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 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+297))) then
        tmp = sqrt((a * (c0 * ((c0 / l) / v))))
    else
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+297)) {
		tmp = Math.sqrt((A * (c0 * ((c0 / l) / V))));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+297):
		tmp = math.sqrt((A * (c0 * ((c0 / l) / V))))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, 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 <= 2e+297))
		tmp = sqrt(Float64(A * Float64(c0 * Float64(Float64(c0 / l) / V))));
	else
		tmp = Float64(c0 * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+297)))
		tmp = sqrt((A * (c0 * ((c0 / l) / V))));
	else
		tmp = c0 * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+297]], $MachinePrecision]], N[Sqrt[N[(A * N[(c0 * N[(N[(c0 / l), $MachinePrecision] / V), $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 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 39.5%

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

      1. add-sqr-sqrt 30.0%

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

      2. sqrt-unprod 30.0%

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

      3. *-commutative 30.0%

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

      4. *-commutative 30.0%

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

      5. swap-sqr 29.4%

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

      6. add-sqr-sqrt 29.4%

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

      7. pow2 29.4%

         \[\leadsto \sqrt{\frac{A}{V \cdot \ell} \cdot \color{blue}{{c0}^{2}}} \]

   4. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   5. Step-by-step derivation

      1. associate-/r* 31.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}} \cdot {c0}^{2}} \]

   6. Simplified 31.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell} \cdot {c0}^{2}}} \]

   7. Taylor expanded in A around 0 32.6%

      \[\leadsto \sqrt{\color{blue}{\frac{A \cdot {c0}^{2}}{V \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 32.6%

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{{c0}^{2}}{V \cdot \ell}}} \]

      2. *-commutative 32.6%

         \[\leadsto \sqrt{A \cdot \frac{{c0}^{2}}{\color{blue}{\ell \cdot V}}} \]

      3. associate-/r* 33.1%

         \[\leadsto \sqrt{A \cdot \color{blue}{\frac{\frac{{c0}^{2}}{\ell}}{V}}} \]

   9. Simplified 33.1%

      \[\leadsto \sqrt{\color{blue}{A \cdot \frac{\frac{{c0}^{2}}{\ell}}{V}}} \]
   10. Step-by-step derivation

       1. unpow2 33.1%

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

       2. associate-/l* 39.9%

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

   11. Applied egg-rr 39.9%

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

       1. associate-/l* 42.2%

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

   13. Applied egg-rr 42.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2e297

   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 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;\sqrt{A \cdot \left(c0 \cdot \frac{\frac{c0}{\ell}}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+297)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && 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 <= 2e+297)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(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 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+297))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+297)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+297):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, 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 <= 2e+297))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+297)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+297]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 2e297 < (/.f64 A (*.f64 V l))

   1. Initial program 39.5%

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

      1. *-commutative 39.5%

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

      2. associate-/l/ 60.4%

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

   3. Simplified 60.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2e297

   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 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

3. Taylor expanded in c0 around 0 69.6%

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

   1. *-commutative 69.6%

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

   2. associate-/r* 76.0%

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

5. Simplified 76.0%

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

6. Final simplification 76.0%

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

Alternative 12: 73.3% 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 69.6%

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

Reproduce

herbie shell --seed 2024166 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))