Henrywood and Agarwal, Equation (3)

Percentage Accurate: 72.8% → 92.6%

Time: 12.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% 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: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-A}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+276}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{-V}}{t_0}}}{\frac{\sqrt{\ell}}{c0_m}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0_m \cdot \frac{t_0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0_m}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- A))))
   (*
    c0_s
    (if (<= (* V l) -5e+276)
      (/ (/ 1.0 (/ (sqrt (- V)) t_0)) (/ (sqrt l) c0_m))
      (if (<= (* V l) -2e-282)
        (* c0_m (/ t_0 (sqrt (* V (- l)))))
        (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+248)))
          (sqrt (* (/ A (/ l c0_m)) (/ (/ 1.0 V) (/ 1.0 c0_m))))
          (* c0_m (/ (sqrt A) (sqrt (* V l))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = sqrt(-A);
	double tmp;
	if ((V * l) <= -5e+276) {
		tmp = (1.0 / (sqrt(-V) / t_0)) / (sqrt(l) / c0_m);
	} else if ((V * l) <= -2e-282) {
		tmp = c0_m * (t_0 / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+248)) {
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-a)
    if ((v * l) <= (-5d+276)) then
        tmp = (1.0d0 / (sqrt(-v) / t_0)) / (sqrt(l) / c0_m)
    else if ((v * l) <= (-2d-282)) then
        tmp = c0_m * (t_0 / sqrt((v * -l)))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 2d+248))) then
        tmp = sqrt(((a / (l / c0_m)) * ((1.0d0 / v) / (1.0d0 / c0_m))))
    else
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = Math.sqrt(-A);
	double tmp;
	if ((V * l) <= -5e+276) {
		tmp = (1.0 / (Math.sqrt(-V) / t_0)) / (Math.sqrt(l) / c0_m);
	} else if ((V * l) <= -2e-282) {
		tmp = c0_m * (t_0 / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+248)) {
		tmp = Math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = math.sqrt(-A)
	tmp = 0
	if (V * l) <= -5e+276:
		tmp = (1.0 / (math.sqrt(-V) / t_0)) / (math.sqrt(l) / c0_m)
	elif (V * l) <= -2e-282:
		tmp = c0_m * (t_0 / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 2e+248):
		tmp = math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))))
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(Float64(-A))
	tmp = 0.0
	if (Float64(V * l) <= -5e+276)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(-V)) / t_0)) / Float64(sqrt(l) / c0_m));
	elseif (Float64(V * l) <= -2e-282)
		tmp = Float64(c0_m * Float64(t_0 / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+248))
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) * Float64(Float64(1.0 / V) / Float64(1.0 / c0_m))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(-A);
	tmp = 0.0;
	if ((V * l) <= -5e+276)
		tmp = (1.0 / (sqrt(-V) / t_0)) / (sqrt(l) / c0_m);
	elseif ((V * l) <= -2e-282)
		tmp = c0_m * (t_0 / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 2e+248)))
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	else
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[(-A)], $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e+276], N[(N[(1.0 / N[(N[Sqrt[(-V)], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-282], N[(c0$95$m * N[(t$95$0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+248]], $MachinePrecision]], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / V), $MachinePrecision] / N[(1.0 / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 42.5%

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

      1. associate-/r* 66.2%

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

      2. sqrt-div 69.5%

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

      3. associate-*r/ 69.6%

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

   3. Applied egg-rr 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/l* 69.3%

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

   5. Simplified 69.3%

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

      1. clear-num 69.4%

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

      2. sqrt-div 69.5%

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

      3. metadata-eval 69.5%

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

   7. Applied egg-rr 69.5%

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

      1. frac-2neg 69.5%

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

      2. sqrt-div 74.8%

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

   9. Applied egg-rr 74.8%

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

   if -5.00000000000000001e276 < (*.f64 V l) < -2e-282

   1. Initial program 85.8%

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

      1. frac-2neg 85.8%

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

      2. sqrt-div 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. distribute-rgt-neg-out 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   5. Simplified 99.3%

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

   if -2e-282 < (*.f64 V l) < 0.0 or 2.00000000000000009e248 < (*.f64 V l)

   1. Initial program 49.0%

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

      1. clear-num 49.0%

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

      2. associate-/r/ 49.0%

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

      3. associate-/r* 48.9%

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

   3. Applied egg-rr 48.9%

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

      1. sqrt-prod 37.2%

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

      2. *-commutative 37.2%

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

      3. *-commutative 37.2%

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

      4. sqrt-div 32.1%

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

      5. associate-*r/ 32.1%

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

      6. sqrt-prod 42.0%

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

      7. div-inv 42.0%

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

      8. associate-/r/ 40.4%

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

      9. add-sqr-sqrt 24.9%

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

      10. sqrt-unprod 21.8%

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

      11. associate-/r/ 21.8%

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

      12. associate-/r/ 21.8%

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

      13. swap-sqr 21.3%

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

   5. Applied egg-rr 37.2%

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

      1. associate-*l/ 35.7%

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

      2. associate-/l* 37.4%

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

   7. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. unpow2 37.4%

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

      3. times-frac 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. div-inv 37.8%

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

       2. *-commutative 37.8%

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

       3. times-frac 42.8%

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

   11. Applied egg-rr 42.8%

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

   if 0.0 < (*.f64 V l) < 2.00000000000000009e248

   1. Initial program 89.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 94.6%

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

      3. associate-/r/ 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+276}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{-V}}{\sqrt{-A}}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 2: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;c0_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0_m \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 2e-305)
      (* c0_m (sqrt (/ (/ A V) l)))
      (if (<= t_0 1e+193) t_0 (* c0_m (pow (* l (/ V A)) -0.5)))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-305) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+193) {
		tmp = t_0;
	} else {
		tmp = c0_m * pow((l * (V / A)), -0.5);
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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_m * sqrt((a / (v * l)))
    if (t_0 <= 2d-305) then
        tmp = c0_m * sqrt(((a / v) / l))
    else if (t_0 <= 1d+193) then
        tmp = t_0
    else
        tmp = c0_m * ((l * (v / a)) ** (-0.5d0))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-305) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+193) {
		tmp = t_0;
	} else {
		tmp = c0_m * Math.pow((l * (V / A)), -0.5);
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-305:
		tmp = c0_m * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+193:
		tmp = t_0
	else:
		tmp = c0_m * math.pow((l * (V / A)), -0.5)
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-305)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+193)
		tmp = t_0;
	else
		tmp = Float64(c0_m * (Float64(l * Float64(V / A)) ^ -0.5));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-305)
		tmp = c0_m * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+193)
		tmp = t_0;
	else
		tmp = c0_m * ((l * (V / A)) ^ -0.5);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 2e-305], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+193], t$95$0, N[(c0$95$m * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999999e-305

   1. Initial program 68.2%

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

      1. associate-/r* 71.5%

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

   3. Simplified 71.5%

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

   if 1.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000007e193

   1. Initial program 98.6%

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

   if 1.00000000000000007e193 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 74.5%

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

      1. clear-num 74.5%

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

      2. associate-/r/ 74.5%

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

      3. associate-/r* 74.5%

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

   3. Applied egg-rr 74.5%

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

      1. associate-*l/ 81.6%

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

      2. associate-*r/ 81.6%

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

      3. associate-*l/ 81.6%

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

      4. *-un-lft-identity 81.6%

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

   5. Applied egg-rr 81.6%

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

      1. div-inv 81.6%

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

      2. associate-*l/ 81.5%

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

      3. sqrt-prod 54.1%

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

      4. associate-*r* 54.1%

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

      5. *-commutative 54.1%

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

      6. sqrt-div 54.1%

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

      7. metadata-eval 54.1%

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

      8. div-inv 54.2%

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

      9. clear-num 54.1%

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

      10. associate-/r* 54.0%

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

      11. clear-num 54.1%

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

      12. sqrt-div 21.9%

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

      13. associate-/r* 21.8%

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

      14. sqrt-prod 37.0%

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

      15. clear-num 37.1%

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

      16. sqrt-undiv 75.4%

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

      17. *-commutative 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. associate-/r/ 75.5%

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

      2. pow1/2 75.5%

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

      3. associate-*r/ 81.6%

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

      4. pow-flip 81.7%

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

      5. metadata-eval 81.7%

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

   9. Applied egg-rr 81.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+193}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{-305} \lor \neg \left(t_0 \leq 5 \cdot 10^{+215}\right):\\ \;\;\;\;c0_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 2e-305) (not (<= t_0 5e+215)))
      (* c0_m (sqrt (/ (/ A V) l)))
      t_0))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-305) || !(t_0 <= 5e+215)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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_m * sqrt((a / (v * l)))
    if ((t_0 <= 2d-305) .or. (.not. (t_0 <= 5d+215))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-305) || !(t_0 <= 5e+215)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 2e-305) or not (t_0 <= 5e+215):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 2e-305) || !(t_0 <= 5e+215))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 2e-305) || ~((t_0 <= 5e+215)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 2e-305], N[Not[LessEqual[t$95$0, 5e+215]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999999e-305 or 5.0000000000000001e215 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.5%

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

      1. associate-/r* 72.8%

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

   3. Simplified 72.8%

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

   if 1.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000001e215

   1. Initial program 98.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-305} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+215}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 4: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;c0_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 2e-305)
      (* c0_m (sqrt (/ (/ A V) l)))
      (if (<= t_0 1e+219) t_0 (/ c0_m (sqrt (* l (/ V A)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-305) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+219) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((l * (V / A)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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_m * sqrt((a / (v * l)))
    if (t_0 <= 2d-305) then
        tmp = c0_m * sqrt(((a / v) / l))
    else if (t_0 <= 1d+219) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((l * (v / a)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-305) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+219) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((l * (V / A)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-305:
		tmp = c0_m * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+219:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((l * (V / A)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-305)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+219)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / A))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-305)
		tmp = c0_m * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+219)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((l * (V / A)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 2e-305], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+219], t$95$0, N[(c0$95$m / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999999e-305

   1. Initial program 68.2%

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

      1. associate-/r* 71.5%

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

   3. Simplified 71.5%

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

   if 1.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999965e218

   1. Initial program 98.7%

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

   if 9.99999999999999965e218 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 70.3%

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

      1. clear-num 70.3%

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

      2. associate-/r/ 70.3%

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

      3. associate-/r* 70.3%

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

   3. Applied egg-rr 70.3%

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

      1. associate-*l/ 78.6%

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

      2. associate-*r/ 78.6%

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

      3. associate-*l/ 78.6%

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

      4. *-un-lft-identity 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. associate-/l/ 70.3%

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

      2. sqrt-undiv 34.9%

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

      3. clear-num 34.9%

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

      4. un-div-inv 34.9%

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

      5. sqrt-undiv 71.4%

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

      6. *-commutative 71.4%

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

   7. Applied egg-rr 71.4%

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

      1. associate-*r/ 78.6%

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

   9. Simplified 78.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+219}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 5: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0_m \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0_m}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -5e+276)
    (/ (* c0_m (sqrt (/ A V))) (sqrt l))
    (if (<= (* V l) -2e-282)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+248)))
        (sqrt (* (/ A (/ l c0_m)) (/ (/ 1.0 V) (/ 1.0 c0_m))))
        (* c0_m (/ (sqrt A) (sqrt (* V l)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+276) {
		tmp = (c0_m * sqrt((A / V))) / sqrt(l);
	} else if ((V * l) <= -2e-282) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+248)) {
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+276)) then
        tmp = (c0_m * sqrt((a / v))) / sqrt(l)
    else if ((v * l) <= (-2d-282)) then
        tmp = c0_m * (sqrt(-a) / sqrt((v * -l)))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 2d+248))) then
        tmp = sqrt(((a / (l / c0_m)) * ((1.0d0 / v) / (1.0d0 / c0_m))))
    else
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+276) {
		tmp = (c0_m * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((V * l) <= -2e-282) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+248)) {
		tmp = Math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -5e+276:
		tmp = (c0_m * math.sqrt((A / V))) / math.sqrt(l)
	elif (V * l) <= -2e-282:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 2e+248):
		tmp = math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))))
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+276)
		tmp = Float64(Float64(c0_m * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(V * l) <= -2e-282)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+248))
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) * Float64(Float64(1.0 / V) / Float64(1.0 / c0_m))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+276)
		tmp = (c0_m * sqrt((A / V))) / sqrt(l);
	elseif ((V * l) <= -2e-282)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 2e+248)))
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	else
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e+276], N[(N[(c0$95$m * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-282], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+248]], $MachinePrecision]], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / V), $MachinePrecision] / N[(1.0 / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 42.5%

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

      1. *-commutative 42.5%

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

      2. associate-/r* 66.2%

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

      3. sqrt-div 69.5%

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

      4. associate-*l/ 69.6%

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

   3. Applied egg-rr 69.6%

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

   if -5.00000000000000001e276 < (*.f64 V l) < -2e-282

   1. Initial program 85.8%

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

      1. frac-2neg 85.8%

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

      2. sqrt-div 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. distribute-rgt-neg-out 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   5. Simplified 99.3%

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

   if -2e-282 < (*.f64 V l) < 0.0 or 2.00000000000000009e248 < (*.f64 V l)

   1. Initial program 49.0%

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

      1. clear-num 49.0%

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

      2. associate-/r/ 49.0%

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

      3. associate-/r* 48.9%

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

   3. Applied egg-rr 48.9%

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

      1. sqrt-prod 37.2%

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

      2. *-commutative 37.2%

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

      3. *-commutative 37.2%

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

      4. sqrt-div 32.1%

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

      5. associate-*r/ 32.1%

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

      6. sqrt-prod 42.0%

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

      7. div-inv 42.0%

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

      8. associate-/r/ 40.4%

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

      9. add-sqr-sqrt 24.9%

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

      10. sqrt-unprod 21.8%

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

      11. associate-/r/ 21.8%

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

      12. associate-/r/ 21.8%

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

      13. swap-sqr 21.3%

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

   5. Applied egg-rr 37.2%

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

      1. associate-*l/ 35.7%

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

      2. associate-/l* 37.4%

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

   7. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. unpow2 37.4%

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

      3. times-frac 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. div-inv 37.8%

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

       2. *-commutative 37.8%

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

       3. times-frac 42.8%

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

   11. Applied egg-rr 42.8%

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

   if 0.0 < (*.f64 V l) < 2.00000000000000009e248

   1. Initial program 89.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 94.6%

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

      3. associate-/r/ 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+276}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 6: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0_m \cdot \frac{\sqrt{-\frac{A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0_m}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (* c0_m (/ (sqrt (- (/ A l))) (sqrt (- V))))
    (if (<= (* V l) -2e-282)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (or (<= (* V l) 0.0) (not (<= (* V l) 2e+248)))
        (sqrt (* (/ A (/ l c0_m)) (/ (/ 1.0 V) (/ 1.0 c0_m))))
        (* c0_m (/ (sqrt A) (sqrt (* V l)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0_m * (sqrt(-(A / l)) / sqrt(-V));
	} else if ((V * l) <= -2e-282) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+248)) {
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	}
	return c0_s * tmp;
}

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0_m * (Math.sqrt(-(A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -2e-282) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 2e+248)) {
		tmp = Math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0_m * (math.sqrt(-(A / l)) / math.sqrt(-V))
	elif (V * l) <= -2e-282:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 2e+248):
		tmp = math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))))
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0_m * Float64(sqrt(Float64(-Float64(A / l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -2e-282)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 2e+248))
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) * Float64(Float64(1.0 / V) / Float64(1.0 / c0_m))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0_m * (sqrt(-(A / l)) / sqrt(-V));
	elseif ((V * l) <= -2e-282)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 2e+248)))
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	else
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0$95$m * N[(N[Sqrt[(-N[(A / l), $MachinePrecision])], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-282], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+248]], $MachinePrecision]], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / V), $MachinePrecision] / N[(1.0 / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 36.0%

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

      1. clear-num 36.0%

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

      2. associate-/r/ 36.0%

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

      3. associate-/r* 36.0%

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

   3. Applied egg-rr 36.0%

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

      1. associate-*l/ 67.4%

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

      2. associate-*r/ 67.5%

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

      3. associate-*l/ 67.5%

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

      4. *-un-lft-identity 67.5%

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

   5. Applied egg-rr 67.5%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 79.1%

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

      3. distribute-neg-frac 79.1%

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

   7. Applied egg-rr 79.1%

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

   if -inf.0 < (*.f64 V l) < -2e-282

   1. Initial program 84.5%

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

      1. frac-2neg 84.5%

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

      2. sqrt-div 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. distribute-rgt-neg-out 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   5. Simplified 99.3%

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

   if -2e-282 < (*.f64 V l) < 0.0 or 2.00000000000000009e248 < (*.f64 V l)

   1. Initial program 49.0%

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

      1. clear-num 49.0%

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

      2. associate-/r/ 49.0%

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

      3. associate-/r* 48.9%

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

   3. Applied egg-rr 48.9%

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

      1. sqrt-prod 37.2%

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

      2. *-commutative 37.2%

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

      3. *-commutative 37.2%

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

      4. sqrt-div 32.1%

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

      5. associate-*r/ 32.1%

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

      6. sqrt-prod 42.0%

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

      7. div-inv 42.0%

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

      8. associate-/r/ 40.4%

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

      9. add-sqr-sqrt 24.9%

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

      10. sqrt-unprod 21.8%

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

      11. associate-/r/ 21.8%

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

      12. associate-/r/ 21.8%

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

      13. swap-sqr 21.3%

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

   5. Applied egg-rr 37.2%

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

      1. associate-*l/ 35.7%

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

      2. associate-/l* 37.4%

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

   7. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. unpow2 37.4%

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

      3. times-frac 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. div-inv 37.8%

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

       2. *-commutative 37.8%

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

       3. times-frac 42.8%

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

   11. Applied egg-rr 42.8%

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

   if 0.0 < (*.f64 V l) < 2.00000000000000009e248

   1. Initial program 89.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/l* 94.6%

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

      3. associate-/r/ 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-\frac{A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+291}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0_m}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0_m \cdot \sqrt{t_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 1e+291)))
      (sqrt (* (/ A (/ l c0_m)) (/ (/ 1.0 V) (/ 1.0 c0_m))))
      (* c0_m (sqrt t_0))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291)) {
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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 <= 1d+291))) then
        tmp = sqrt(((a / (l / c0_m)) * ((1.0d0 / v) / (1.0d0 / c0_m))))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291)) {
		tmp = Math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 1e+291):
		tmp = math.sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291))
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) * Float64(Float64(1.0 / V) / Float64(1.0 / c0_m))));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 1e+291)))
		tmp = sqrt(((A / (l / c0_m)) * ((1.0 / V) / (1.0 / c0_m))));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+291]], $MachinePrecision]], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / V), $MachinePrecision] / N[(1.0 / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 9.9999999999999996e290 < (/.f64 A (*.f64 V l))

   1. Initial program 42.6%

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

      1. clear-num 42.6%

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

      2. associate-/r/ 42.6%

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

      3. associate-/r* 42.6%

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

   3. Applied egg-rr 42.6%

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

      1. sqrt-prod 30.1%

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

      2. *-commutative 30.1%

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

      3. *-commutative 30.1%

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

      4. sqrt-div 23.7%

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

      5. associate-*r/ 23.7%

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

      6. sqrt-prod 47.4%

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

      7. div-inv 47.4%

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

      8. associate-/r/ 46.5%

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

      9. add-sqr-sqrt 29.9%

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

      10. sqrt-unprod 25.3%

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

      11. associate-/r/ 25.3%

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

      12. associate-/r/ 25.4%

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

      13. swap-sqr 24.1%

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

   5. Applied egg-rr 33.9%

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

      1. associate-*l/ 33.1%

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

      2. associate-/l* 34.0%

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

   7. Simplified 34.0%

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

      1. *-un-lft-identity 34.0%

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

      2. unpow2 34.0%

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

      3. times-frac 36.2%

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

   9. Applied egg-rr 36.2%

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

       1. div-inv 36.1%

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

       2. *-commutative 36.1%

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

       3. times-frac 41.6%

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

   11. Applied egg-rr 41.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e290

   1. Initial program 99.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+291}\right):\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{c0}} \cdot \frac{\frac{1}{V}}{\frac{1}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 8: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+291}\right):\\ \;\;\;\;\sqrt{\frac{c0_m}{\ell} \cdot \left(c0_m \cdot \frac{A}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0_m \cdot \sqrt{t_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 1e+291)))
      (sqrt (* (/ c0_m l) (* c0_m (/ A V))))
      (* c0_m (sqrt t_0))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291)) {
		tmp = sqrt(((c0_m / l) * (c0_m * (A / V))));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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 <= 1d+291))) then
        tmp = sqrt(((c0_m / l) * (c0_m * (a / v))))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291)) {
		tmp = Math.sqrt(((c0_m / l) * (c0_m * (A / V))));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 1e+291):
		tmp = math.sqrt(((c0_m / l) * (c0_m * (A / V))))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1e+291))
		tmp = sqrt(Float64(Float64(c0_m / l) * Float64(c0_m * Float64(A / V))));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 1e+291)))
		tmp = sqrt(((c0_m / l) * (c0_m * (A / V))));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+291]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m * N[(A / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 9.9999999999999996e290 < (/.f64 A (*.f64 V l))

   1. Initial program 42.6%

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

      1. clear-num 42.6%

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

      2. associate-/r/ 42.6%

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

      3. associate-/r* 42.6%

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

   3. Applied egg-rr 42.6%

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

      1. sqrt-prod 30.1%

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

      2. *-commutative 30.1%

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

      3. *-commutative 30.1%

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

      4. sqrt-div 23.7%

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

      5. associate-*r/ 23.7%

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

      6. sqrt-prod 47.4%

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

      7. div-inv 47.4%

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

      8. associate-/r/ 46.5%

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

      9. add-sqr-sqrt 29.9%

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

      10. sqrt-unprod 25.3%

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

      11. associate-/r/ 25.3%

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

      12. associate-/r/ 25.4%

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

      13. swap-sqr 24.1%

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

   5. Applied egg-rr 33.9%

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

      1. associate-*l/ 33.1%

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

      2. associate-/l* 34.0%

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

   7. Simplified 34.0%

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

      1. *-un-lft-identity 34.0%

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

      2. unpow2 34.0%

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

      3. times-frac 36.2%

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

   9. Applied egg-rr 36.2%

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

       1. *-un-lft-identity 36.2%

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

       2. *-commutative 36.2%

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

       3. times-frac 36.3%

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

       4. clear-num 36.3%

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

       5. div-inv 36.3%

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

       6. clear-num 36.3%

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

       7. /-rgt-identity 36.3%

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

   11. Applied egg-rr 36.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e290

   1. Initial program 99.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+291}\right):\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \left(c0 \cdot \frac{A}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 9: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{\frac{1}{c0_m}} \cdot \frac{\frac{1}{V}}{\frac{\ell}{c0_m}}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;c0_m \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0_m}{\ell} \cdot \left(c0_m \cdot \frac{A}{V}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (sqrt (* (/ A (/ 1.0 c0_m)) (/ (/ 1.0 V) (/ l c0_m))))
      (if (<= t_0 1e+291)
        (* c0_m (sqrt t_0))
        (sqrt (* (/ c0_m l) (* c0_m (/ A V)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / (1.0 / c0_m)) * ((1.0 / V) / (l / c0_m))));
	} else if (t_0 <= 1e+291) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt(((c0_m / l) * (c0_m * (A / V))));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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 = sqrt(((a / (1.0d0 / c0_m)) * ((1.0d0 / v) / (l / c0_m))))
    else if (t_0 <= 1d+291) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt(((c0_m / l) * (c0_m * (a / v))))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / (1.0 / c0_m)) * ((1.0 / V) / (l / c0_m))));
	} else if (t_0 <= 1e+291) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((c0_m / l) * (c0_m * (A / V))));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / (1.0 / c0_m)) * ((1.0 / V) / (l / c0_m))))
	elif t_0 <= 1e+291:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((c0_m / l) * (c0_m * (A / V))))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(A / Float64(1.0 / c0_m)) * Float64(Float64(1.0 / V) / Float64(l / c0_m))));
	elseif (t_0 <= 1e+291)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(c0_m / l) * Float64(c0_m * Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / (1.0 / c0_m)) * ((1.0 / V) / (l / c0_m))));
	elseif (t_0 <= 1e+291)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt(((c0_m / l) * (c0_m * (A / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(A / N[(1.0 / c0$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / V), $MachinePrecision] / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+291], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m * N[(A / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.9%

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

      1. clear-num 31.9%

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

      2. associate-/r/ 31.9%

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

      3. associate-/r* 31.9%

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

   3. Applied egg-rr 31.9%

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

      1. sqrt-prod 32.4%

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

      2. *-commutative 32.4%

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

      3. *-commutative 32.4%

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

      4. sqrt-div 24.8%

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

      5. associate-*r/ 24.8%

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

      6. sqrt-prod 47.3%

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

      7. div-inv 47.3%

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

      8. associate-/r/ 45.7%

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

      9. add-sqr-sqrt 32.3%

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

      10. sqrt-unprod 27.1%

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

      11. associate-/r/ 27.1%

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

      12. associate-/r/ 27.2%

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

      13. swap-sqr 24.9%

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

   5. Applied egg-rr 34.0%

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

      1. associate-*l/ 34.1%

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

      2. associate-/l* 34.1%

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

   7. Simplified 34.1%

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

      1. *-un-lft-identity 34.1%

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

      2. unpow2 34.1%

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

      3. times-frac 37.9%

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

   9. Applied egg-rr 37.9%

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

       1. div-inv 37.9%

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

       2. times-frac 43.0%

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

   11. Applied egg-rr 43.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e290

   1. Initial program 99.3%

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

   if 9.9999999999999996e290 < (/.f64 A (*.f64 V l))

   1. Initial program 55.5%

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

      1. clear-num 55.5%

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

      2. associate-/r/ 55.5%

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

      3. associate-/r* 55.5%

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

   3. Applied egg-rr 55.5%

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

      1. sqrt-prod 27.4%

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

      2. *-commutative 27.4%

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

      3. *-commutative 27.4%

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

      4. sqrt-div 22.4%

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

      5. associate-*r/ 22.4%

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

      6. sqrt-prod 47.6%

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

      7. div-inv 47.6%

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

      8. associate-/r/ 47.5%

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

      9. add-sqr-sqrt 26.9%

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

      10. sqrt-unprod 23.2%

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

      11. associate-/r/ 23.2%

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

      12. associate-/r/ 23.2%

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

      13. swap-sqr 23.1%

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

   5. Applied egg-rr 33.8%

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

      1. associate-*l/ 31.9%

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

      2. associate-/l* 33.9%

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

   7. Simplified 33.9%

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

      1. *-un-lft-identity 33.9%

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

      2. unpow2 33.9%

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

      3. times-frac 34.0%

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

   9. Applied egg-rr 34.0%

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

       1. *-un-lft-identity 34.0%

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

       2. *-commutative 34.0%

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

       3. times-frac 34.1%

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

       4. clear-num 34.1%

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

       5. div-inv 34.1%

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

       6. clear-num 34.1%

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

       7. /-rgt-identity 34.1%

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

   11. Applied egg-rr 34.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{\frac{1}{c0}} \cdot \frac{\frac{1}{V}}{\frac{\ell}{c0}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \left(c0 \cdot \frac{A}{V}\right)}\\ \end{array} \]

Alternative 10: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0_s \cdot \left(c0_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\right) \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (* c0_s (* c0_m (sqrt (/ A (* V l))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	return c0_s * (c0_m * sqrt((A / (V * l))));
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0_s * (c0_m * sqrt((a / (v * l))))
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	return c0_s * (c0_m * Math.sqrt((A / (V * l))));
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	return c0_s * (c0_m * math.sqrt((A / (V * l))))

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	return Float64(c0_s * Float64(c0_m * sqrt(Float64(A / Float64(V * l)))))
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp = code(c0_s, c0_m, A, V, l)
	tmp = c0_s * (c0_m * sqrt((A / (V * l))));
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

2. Final simplification 75.4%

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

Reproduce

herbie shell --seed 2023316 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))