Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.9% → 90.7%

Time: 10.4s

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: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{t\_0}{\sqrt{-V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{{\left(\frac{V}{A}\right)}^{-1}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

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 (sqrt (- A))))
   (if (<= (* V l) (- INFINITY))
     (* (/ t_0 (sqrt (- V))) (/ c0 (sqrt l)))
     (if (<= (* V l) -5e-287)
       (* c0 (/ t_0 (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (* c0 (sqrt (/ (pow (/ V A) -1.0) l)))
         (/ c0 (/ (sqrt (* V l)) (sqrt A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(-A);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (t_0 / sqrt(-V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -5e-287) {
		tmp = c0 * (t_0 / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt((pow((V / A), -1.0) / l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(-A);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.sqrt(-V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-287) {
		tmp = c0 * (t_0 / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt((Math.pow((V / A), -1.0) / l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(-A)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (t_0 / math.sqrt(-V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -5e-287:
		tmp = c0 * (t_0 / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt((math.pow((V / A), -1.0) / l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(-A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 / sqrt(Float64(-V))) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-287)
		tmp = Float64(c0 * Float64(t_0 / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64((Float64(V / A) ^ -1.0) / l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(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 = sqrt(-A);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (t_0 / sqrt(-V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -5e-287)
		tmp = c0 * (t_0 / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt((((V / A) ^ -1.0) / l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(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[Sqrt[(-A)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(t$95$0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-287], N[(c0 * N[(t$95$0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[Power[N[(V / A), $MachinePrecision], -1.0], $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 40.3%

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

      1. associate-/r* 64.2%

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

      2. sqrt-div 44.0%

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

      3. associate-*r/ 44.0%

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

   4. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. associate-/l* 44.0%

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

   6. Simplified 44.0%

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

      1. frac-2neg 44.0%

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

      2. sqrt-div 49.9%

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

   8. Applied egg-rr 49.9%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000025e-287

   1. Initial program 81.8%

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

      1. frac-2neg 81.8%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -5.00000000000000025e-287 < (*.f64 V l) < -0.0

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/l/ 72.1%

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

   3. Simplified 72.1%

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

      1. clear-num 72.1%

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

      2. inv-pow 72.1%

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

   6. Applied egg-rr 72.1%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.8%

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

   3. Taylor expanded in c0 around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-/r* 81.7%

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

   5. Simplified 81.7%

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

      1. div-inv 81.7%

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

      2. associate-*l/ 73.9%

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

      3. div-inv 73.9%

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

      4. clear-num 73.4%

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

      5. unpow-1 73.4%

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

      6. unpow-1 73.4%

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

      7. clear-num 73.9%

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

      8. clear-num 73.8%

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

      9. sqrt-div 73.8%

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

      10. metadata-eval 73.8%

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

      11. associate-/r/ 81.3%

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

      12. *-commutative 81.3%

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

      13. associate-*l/ 81.4%

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

      14. *-un-lft-identity 81.4%

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

      15. *-commutative 81.4%

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

      16. associate-/r/ 73.9%

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

      17. div-inv 73.4%

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

      18. clear-num 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. associate-*r/ 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-/l* 81.4%

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

   9. Simplified 81.4%

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

       1. associate-*r/ 84.6%

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

       2. sqrt-div 93.7%

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

       3. *-commutative 93.7%

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

   11. Applied egg-rr 93.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{{\left(\frac{V}{A}\right)}^{-1}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-191} \lor \neg \left(t\_0 \leq 10^{+282}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 2e-191) (not (<= t_0 1e+282)))
     (/ 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 <= 2e-191) || !(t_0 <= 1e+282)) {
		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 <= 2d-191) .or. (.not. (t_0 <= 1d+282))) 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 <= 2e-191) || !(t_0 <= 1e+282)) {
		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 <= 2e-191) or not (t_0 <= 1e+282):
		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 <= 2e-191) || !(t_0 <= 1e+282))
		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 <= 2e-191) || ~((t_0 <= 1e+282)))
		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, 2e-191], N[Not[LessEqual[t$95$0, 1e+282]], $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)))) < 2e-191 or 1.00000000000000003e282 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.5%

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

   3. Taylor expanded in c0 around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r* 73.2%

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

   5. Simplified 73.2%

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

      1. div-inv 73.2%

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

      2. associate-*l/ 69.3%

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

      3. div-inv 69.3%

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

      4. clear-num 68.5%

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

      5. unpow-1 68.5%

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

      6. unpow-1 68.5%

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

      7. clear-num 69.3%

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

      8. clear-num 69.3%

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

      9. sqrt-div 70.0%

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

      10. metadata-eval 70.0%

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

      11. associate-/r/ 74.0%

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

      12. *-commutative 74.0%

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

      13. associate-*l/ 74.1%

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

      14. *-un-lft-identity 74.1%

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

      15. *-commutative 74.1%

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

      16. associate-/r/ 70.1%

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

      17. div-inv 69.3%

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

      18. clear-num 69.5%

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

   7. Applied egg-rr 69.5%

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

      1. associate-*r/ 66.1%

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

      2. *-commutative 66.1%

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

      3. associate-/l* 74.1%

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

   9. Simplified 74.1%

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

   if 2e-191 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000003e282

   1. Initial program 98.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-191} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+282}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+275}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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+275)))
     (* 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+275)) {
		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+275))) 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+275)) {
		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+275):
		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+275))
		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+275)))
		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+275]], $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 5.0000000000000003e275 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/l/ 68.5%

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

   3. Simplified 68.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000003e275

   1. Initial program 98.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+275}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 10^{-200}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 1e-200)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+275) t_0 (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-200) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+275) {
		tmp = t_0;
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 1d-200) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+275) then
        tmp = t_0
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-200) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+275) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 1e-200:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+275:
		tmp = t_0
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 1e-200)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+275)
		tmp = t_0;
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 1e-200)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+275)
		tmp = t_0;
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-200], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+275], t$95$0, N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999998e-201

   1. Initial program 67.6%

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

   3. Taylor expanded in c0 around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-/r* 73.9%

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

   5. Simplified 73.9%

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

   if 9.9999999999999998e-201 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000003e275

   1. Initial program 98.3%

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

   if 5.0000000000000003e275 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

      2. associate-/l/ 64.5%

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

   3. Simplified 64.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-200}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{{\left(\frac{V}{A}\right)}^{-1}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

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) (- INFINITY))
   (/ (/ c0 (sqrt l)) (sqrt (/ V A)))
   (if (<= (* V l) -5e-287)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (sqrt (/ (pow (/ V A) -1.0) l)))
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))))))

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(l)) / sqrt((V / A));
	} else if ((V * l) <= -5e-287) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt((pow((V / A), -1.0) / l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	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(l)) / Math.sqrt((V / A));
	} else if ((V * l) <= -5e-287) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt((Math.pow((V / A), -1.0) / l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	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(l)) / math.sqrt((V / A))
	elif (V * l) <= -5e-287:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt((math.pow((V / A), -1.0) / l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(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) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) / sqrt(Float64(V / A)));
	elseif (Float64(V * l) <= -5e-287)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64((Float64(V / A) ^ -1.0) / l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(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) <= -Inf)
		tmp = (c0 / sqrt(l)) / sqrt((V / A));
	elseif ((V * l) <= -5e-287)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt((((V / A) ^ -1.0) / l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(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], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-287], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[Power[N[(V / A), $MachinePrecision], -1.0], $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 40.3%

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

   3. Taylor expanded in c0 around 0 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-/r* 64.2%

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

   5. Simplified 64.2%

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

      1. associate-/l/ 40.3%

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

      2. sqrt-div 0.0%

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

      3. *-commutative 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. sqrt-div 40.3%

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

      3. *-commutative 40.3%

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

      4. associate-*r/ 64.2%

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

      5. pow1/2 64.2%

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

      6. associate-*r/ 40.3%

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

      7. *-commutative 40.3%

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

      8. associate-*r/ 64.2%

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

      9. pow1/2 64.2%

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

      10. associate-*l/ 64.2%

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

      11. sqrt-prod 43.9%

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

      12. *-un-lft-identity 43.9%

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

      13. associate-/r* 44.0%

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

   9. Applied egg-rr 44.0%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000025e-287

   1. Initial program 81.8%

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

      1. frac-2neg 81.8%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -5.00000000000000025e-287 < (*.f64 V l) < -0.0

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/l/ 72.1%

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

   3. Simplified 72.1%

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

      1. clear-num 72.1%

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

      2. inv-pow 72.1%

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

   6. Applied egg-rr 72.1%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.8%

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

   3. Taylor expanded in c0 around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-/r* 81.7%

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

   5. Simplified 81.7%

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

      1. div-inv 81.7%

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

      2. associate-*l/ 73.9%

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

      3. div-inv 73.9%

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

      4. clear-num 73.4%

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

      5. unpow-1 73.4%

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

      6. unpow-1 73.4%

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

      7. clear-num 73.9%

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

      8. clear-num 73.8%

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

      9. sqrt-div 73.8%

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

      10. metadata-eval 73.8%

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

      11. associate-/r/ 81.3%

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

      12. *-commutative 81.3%

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

      13. associate-*l/ 81.4%

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

      14. *-un-lft-identity 81.4%

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

      15. *-commutative 81.4%

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

      16. associate-/r/ 73.9%

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

      17. div-inv 73.4%

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

      18. clear-num 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. associate-*r/ 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-/l* 81.4%

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

   9. Simplified 81.4%

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

       1. associate-*r/ 84.6%

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

       2. sqrt-div 93.7%

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

       3. *-commutative 93.7%

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

   11. Applied egg-rr 93.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{{\left(\frac{V}{A}\right)}^{-1}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.3%

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

      1. associate-/r* 73.5%

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

      2. sqrt-div 44.4%

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

      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. associate-/l* 44.4%

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

   6. Simplified 44.4%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.8%

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

   3. Taylor expanded in c0 around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-/r* 81.7%

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

   5. Simplified 81.7%

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

      1. div-inv 81.7%

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

      2. associate-*l/ 73.9%

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

      3. div-inv 73.9%

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

      4. clear-num 73.4%

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

      5. unpow-1 73.4%

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

      6. unpow-1 73.4%

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

      7. clear-num 73.9%

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

      8. clear-num 73.8%

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

      9. sqrt-div 73.8%

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

      10. metadata-eval 73.8%

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

      11. associate-/r/ 81.3%

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

      12. *-commutative 81.3%

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

      13. associate-*l/ 81.4%

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

      14. *-un-lft-identity 81.4%

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

      15. *-commutative 81.4%

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

      16. associate-/r/ 73.9%

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

      17. div-inv 73.4%

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

      18. clear-num 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. associate-*r/ 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-/l* 81.4%

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

   9. Simplified 81.4%

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

       1. associate-*r/ 84.6%

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

       2. sqrt-div 93.7%

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

       3. *-commutative 93.7%

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

   11. Applied egg-rr 93.7%

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

4. Final simplification 67.1%

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

Alternative 7: 84.2% accurate, 0.5× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* 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) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} 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) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    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) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} 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:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	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)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	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], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.3%

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

      1. associate-/r* 73.5%

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

      2. sqrt-div 44.4%

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

      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. associate-/l* 44.4%

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

   6. Simplified 44.4%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.8%

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

      1. sqrt-div 93.6%

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

      2. associate-*r/ 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. associate-/l* 93.6%

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

   6. Simplified 93.6%

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

Alternative 8: 80.6% accurate, 0.5× speedup

Math

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

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 (<= (* V l) 0.0)
   (/ 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) {
		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) 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) {
		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:
		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)
		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)
		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[LessEqual[N[(V * l), $MachinePrecision], 0.0], 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

   1. Initial program 65.3%

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

   3. Taylor expanded in c0 around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-/r* 73.9%

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

   5. Simplified 73.9%

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

      1. div-inv 73.9%

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

      2. associate-*l/ 73.5%

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

      3. div-inv 73.5%

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

      4. clear-num 72.8%

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

      5. unpow-1 72.8%

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

      6. unpow-1 72.8%

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

      7. clear-num 73.5%

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

      8. clear-num 73.5%

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

      9. sqrt-div 74.5%

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

      10. metadata-eval 74.5%

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

      11. associate-/r/ 74.8%

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

      12. *-commutative 74.8%

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

      13. associate-*l/ 75.0%

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

      14. *-un-lft-identity 75.0%

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

      15. *-commutative 75.0%

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

      16. associate-/r/ 74.6%

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

      17. div-inv 74.0%

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

      18. clear-num 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. associate-*r/ 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-/l* 75.0%

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

   9. Simplified 75.0%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.8%

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

      1. sqrt-div 93.6%

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

      2. associate-*r/ 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. associate-/l* 93.6%

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

   6. Simplified 93.6%

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

Alternative 9: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-319} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \left(c0 \cdot \frac{1}{V}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 2e-319) (not (<= t_0 5e+299)))
     (sqrt (* (/ A l) (* c0 (* c0 (/ 1.0 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 <= 2e-319) || !(t_0 <= 5e+299)) {
		tmp = sqrt(((A / l) * (c0 * (c0 * (1.0 / 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 <= 2d-319) .or. (.not. (t_0 <= 5d+299))) then
        tmp = sqrt(((a / l) * (c0 * (c0 * (1.0d0 / 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 <= 2e-319) || !(t_0 <= 5e+299)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 * (1.0 / 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 <= 2e-319) or not (t_0 <= 5e+299):
		tmp = math.sqrt(((A / l) * (c0 * (c0 * (1.0 / 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 <= 2e-319) || !(t_0 <= 5e+299))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 * Float64(1.0 / 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 <= 2e-319) || ~((t_0 <= 5e+299)))
		tmp = sqrt(((A / l) * (c0 * (c0 * (1.0 / 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, 2e-319], N[Not[LessEqual[t$95$0, 5e+299]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 * N[(1.0 / V), $MachinePrecision]), $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)) < 1.99998e-319 or 5.0000000000000003e299 < (/.f64 A (*.f64 V l))

   1. Initial program 36.1%

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

      1. add-sqr-sqrt 25.5%

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

      2. sqrt-unprod 25.6%

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

      3. *-commutative 25.6%

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

      4. *-commutative 25.6%

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

      5. swap-sqr 25.1%

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

      6. add-sqr-sqrt 25.1%

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

      7. pow2 25.1%

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

   4. Applied egg-rr 25.1%

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

      1. associate-*l/ 29.2%

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

      2. *-commutative 29.2%

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

      3. times-frac 35.3%

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

   6. Simplified 35.3%

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

      1. div-inv 35.3%

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

      2. unpow2 35.3%

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

      3. associate-*l* 39.0%

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

   8. Applied egg-rr 39.0%

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

   if 1.99998e-319 < (/.f64 A (*.f64 V l)) < 5.0000000000000003e299

   1. Initial program 98.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-319} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \left(c0 \cdot \frac{1}{V}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+277}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (* (/ A l) (/ 1.0 V))))
     (if (<= t_0 4e+277) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

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) {
		tmp = c0 * sqrt(((A / l) * (1.0 / V)));
	} else if (t_0 <= 4e+277) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (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) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) * (1.0d0 / v)))
    else if (t_0 <= 4d+277) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) * (1.0 / V)));
	} else if (t_0 <= 4e+277) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	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:
		tmp = c0 * math.sqrt(((A / l) * (1.0 / V)))
	elif t_0 <= 4e+277:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	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)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) * Float64(1.0 / V))));
	elseif (t_0 <= 4e+277)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) * (1.0 / V)));
	elseif (t_0 <= 4e+277)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(1.0 / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+277], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.2%

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

      1. *-un-lft-identity 39.2%

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

      2. times-frac 54.7%

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

   4. Applied egg-rr 54.7%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000001e277

   1. Initial program 98.5%

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

   if 4.00000000000000001e277 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

   3. Taylor expanded in c0 around 0 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r* 57.9%

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

   5. Simplified 57.9%

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

      1. div-inv 57.9%

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

      2. associate-*l/ 56.3%

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

      3. div-inv 56.3%

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

      4. clear-num 56.3%

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

      5. unpow-1 56.3%

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

      6. unpow-1 56.3%

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

      7. clear-num 56.3%

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

      8. clear-num 56.3%

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

      9. sqrt-div 58.7%

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

      10. metadata-eval 58.7%

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

      11. associate-/r/ 60.3%

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

      12. *-commutative 60.3%

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

      13. associate-*l/ 60.4%

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

      14. *-un-lft-identity 60.4%

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

      15. *-commutative 60.4%

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

      16. associate-/r/ 58.7%

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

      17. div-inv 58.7%

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

      18. clear-num 58.7%

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

   7. Applied egg-rr 58.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+277}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+277}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+277) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

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) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+277) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (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) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+277) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+277) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	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:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+277:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	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)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+277)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+277)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+277], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.2%

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

      1. *-commutative 39.2%

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

      2. associate-/l/ 54.7%

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

   3. Simplified 54.7%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000001e277

   1. Initial program 98.5%

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

   if 4.00000000000000001e277 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

   3. Taylor expanded in c0 around 0 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r* 57.9%

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

   5. Simplified 57.9%

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

      1. div-inv 57.9%

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

      2. associate-*l/ 56.3%

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

      3. div-inv 56.3%

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

      4. clear-num 56.3%

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

      5. unpow-1 56.3%

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

      6. unpow-1 56.3%

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

      7. clear-num 56.3%

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

      8. clear-num 56.3%

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

      9. sqrt-div 58.7%

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

      10. metadata-eval 58.7%

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

      11. associate-/r/ 60.3%

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

      12. *-commutative 60.3%

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

      13. associate-*l/ 60.4%

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

      14. *-un-lft-identity 60.4%

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

      15. *-commutative 60.4%

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

      16. associate-/r/ 58.7%

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

      17. div-inv 58.7%

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

      18. clear-num 58.7%

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

   7. Applied egg-rr 58.7%

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

Alternative 12: 73.9% 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 74.3%

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

Reproduce

herbie shell --seed 2024151 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))