Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 94.0%

Time: 18.7s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0\_m}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{+290}:\\ \;\;\;\;c0\_m \cdot \left(\frac{\sqrt{A}}{\sqrt{V \cdot \sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(A \cdot c0\_m\right) \cdot \frac{c0\_m}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= A -1e-310)
    (/ (* (/ (sqrt (- A)) (sqrt (- V))) c0_m) (sqrt l))
    (if (<= A 5.2e+290)
      (*
       c0_m
       (* (/ (sqrt A) (sqrt (* V (cbrt l)))) (sqrt (pow (cbrt l) -2.0))))
      (sqrt (/ (* (* A c0_m) (/ c0_m 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 (A <= -1e-310) {
		tmp = ((sqrt(-A) / sqrt(-V)) * c0_m) / sqrt(l);
	} else if (A <= 5.2e+290) {
		tmp = c0_m * ((sqrt(A) / sqrt((V * cbrt(l)))) * sqrt(pow(cbrt(l), -2.0)));
	} else {
		tmp = sqrt((((A * c0_m) * (c0_m / 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 (A <= -1e-310) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) * c0_m) / Math.sqrt(l);
	} else if (A <= 5.2e+290) {
		tmp = c0_m * ((Math.sqrt(A) / Math.sqrt((V * Math.cbrt(l)))) * Math.sqrt(Math.pow(Math.cbrt(l), -2.0)));
	} else {
		tmp = Math.sqrt((((A * c0_m) * (c0_m / 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 (A <= -1e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * c0_m) / sqrt(l));
	elseif (A <= 5.2e+290)
		tmp = Float64(c0_m * Float64(Float64(sqrt(A) / sqrt(Float64(V * cbrt(l)))) * sqrt((cbrt(l) ^ -2.0))));
	else
		tmp = sqrt(Float64(Float64(Float64(A * c0_m) * Float64(c0_m / V)) / l));
	end
	return Float64(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[A, -1e-310], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0$95$m), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 5.2e+290], N[(c0$95$m * N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(A * c0$95$m), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if A < -9.999999999999969e-311

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-/r* 77.5%

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

      3. sqrt-div 44.4%

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

      4. associate-*l/ 44.5%

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

   4. Applied egg-rr 44.5%

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

      1. frac-2neg 44.5%

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

      2. sqrt-div 54.5%

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

   6. Applied egg-rr 54.5%

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

   if -9.999999999999969e-311 < A < 5.19999999999999997e290

   1. Initial program 72.1%

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

      1. associate-/r* 69.9%

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

      2. div-inv 69.8%

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

      3. add-cube-cbrt 69.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 74.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 74.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 74.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 73.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 72.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. div-inv 72.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \sqrt[3]{\ell}} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. sqrt-prod 77.3%

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)} \]

      3. associate-/r* 74.8%

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}} \cdot \sqrt{\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right) \]

      4. pow-flip 74.9%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}}}\right) \]

      5. metadata-eval 74.9%

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}}\right) \]

   8. Applied egg-rr 74.9%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)} \]
   9. Step-by-step derivation

      1. associate-/l/ 77.3%

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot V}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \]

      2. sqrt-div 93.6%

         \[\leadsto c0 \cdot \left(\color{blue}{\frac{\sqrt{A}}{\sqrt{\sqrt[3]{\ell} \cdot V}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \]

   10. Applied egg-rr 93.6%

       \[\leadsto c0 \cdot \left(\color{blue}{\frac{\sqrt{A}}{\sqrt{\sqrt[3]{\ell} \cdot V}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \]

   if 5.19999999999999997e290 < A

   1. Initial program 51.7%

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

      1. associate-/r* 51.6%

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

      2. div-inv 51.6%

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

      3. add-cube-cbrt 51.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 67.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 67.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 67.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 51.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 51.3%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 51.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 51.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 51.6%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 51.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 17.5%

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

      1. associate-*l/ 18.1%

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

      2. associate-/l* 18.1%

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

   10. Simplified 18.1%

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

       1. unpow2 18.1%

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

       2. associate-/l* 18.2%

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

   12. Applied egg-rr 18.2%

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

       1. associate-*r* 17.9%

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

       2. associate-/r* 18.3%

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

       3. associate-*r/ 34.4%

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

   14. Applied egg-rr 34.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \left(\frac{\sqrt{A}}{\sqrt{V \cdot \sqrt[3]{\ell}}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(A \cdot c0\right) \cdot \frac{c0}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\frac{\ell}{c0\_m}}}{\frac{V}{c0\_m}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{V}{c0\_m} \cdot \frac{\frac{\ell}{c0\_m}}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0)
      (sqrt (/ (/ A (/ l c0_m)) (/ V c0_m)))
      (if (<= t_0 2e+274)
        t_0
        (/ 1.0 (sqrt (* (/ V c0_m) (/ (/ l c0_m) 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 <= 0.0) {
		tmp = sqrt(((A / (l / c0_m)) / (V / c0_m)));
	} else if (t_0 <= 2e+274) {
		tmp = t_0;
	} else {
		tmp = 1.0 / sqrt(((V / c0_m) * ((l / c0_m) / 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 <= 0.0d0) then
        tmp = sqrt(((a / (l / c0_m)) / (v / c0_m)))
    else if (t_0 <= 2d+274) then
        tmp = t_0
    else
        tmp = 1.0d0 / sqrt(((v / c0_m) * ((l / c0_m) / 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 <= 0.0) {
		tmp = Math.sqrt(((A / (l / c0_m)) / (V / c0_m)));
	} else if (t_0 <= 2e+274) {
		tmp = t_0;
	} else {
		tmp = 1.0 / Math.sqrt(((V / c0_m) * ((l / c0_m) / 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 <= 0.0:
		tmp = math.sqrt(((A / (l / c0_m)) / (V / c0_m)))
	elif t_0 <= 2e+274:
		tmp = t_0
	else:
		tmp = 1.0 / math.sqrt(((V / c0_m) * ((l / c0_m) / 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 <= 0.0)
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) / Float64(V / c0_m)));
	elseif (t_0 <= 2e+274)
		tmp = t_0;
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(V / c0_m) * Float64(Float64(l / c0_m) / 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 <= 0.0)
		tmp = sqrt(((A / (l / c0_m)) / (V / c0_m)));
	elseif (t_0 <= 2e+274)
		tmp = t_0;
	else
		tmp = 1.0 / sqrt(((V / c0_m) * ((l / c0_m) / 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, 0.0], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2e+274], t$95$0, N[(1.0 / N[Sqrt[N[(N[(V / c0$95$m), $MachinePrecision] * N[(N[(l / c0$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

      1. associate-/r* 71.5%

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

      2. div-inv 71.6%

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

      3. add-cube-cbrt 71.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 71.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 71.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 72.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 17.2%

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

      1. associate-*l/ 18.4%

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

      2. associate-/l* 18.5%

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

   10. Simplified 18.5%

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

       1. unpow2 18.5%

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

       2. *-commutative 18.5%

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

       3. times-frac 23.3%

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

   12. Applied egg-rr 23.3%

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

       1. *-commutative 23.3%

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

       2. clear-num 23.3%

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

       3. frac-times 22.7%

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

       4. *-un-lft-identity 22.7%

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

   14. Applied egg-rr 22.7%

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

       1. associate-*r/ 22.7%

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

       2. *-commutative 22.7%

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

       3. associate-/r* 23.2%

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

       4. clear-num 23.2%

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

       5. associate-/l/ 23.8%

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

       6. clear-num 23.8%

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

   16. Applied egg-rr 23.8%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999984e274

   1. Initial program 99.1%

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

   if 1.99999999999999984e274 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 44.2%

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

      1. associate-/r* 44.5%

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

      2. div-inv 44.5%

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

      3. add-cube-cbrt 44.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 48.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 48.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 48.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 44.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 44.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 44.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 44.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 44.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 44.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 35.8%

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

      1. associate-*l/ 36.3%

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

      2. associate-/l* 36.3%

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

   10. Simplified 36.3%

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

       1. unpow2 36.3%

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

       2. associate-/l* 44.1%

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

   12. Applied egg-rr 44.1%

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

       1. associate-*r* 48.0%

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

       2. clear-num 48.0%

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

       3. un-div-inv 48.0%

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

   14. Applied egg-rr 48.0%

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

       1. clear-num 48.0%

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

       2. sqrt-div 48.0%

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

       3. metadata-eval 48.0%

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

       4. associate-/l* 60.4%

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

       5. *-commutative 60.4%

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

       6. times-frac 68.3%

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

   16. Applied egg-rr 68.3%

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

Alternative 3: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+278}\right):\\ \;\;\;\;\sqrt{\frac{\frac{A}{\frac{\ell}{c0\_m}}}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0) (not (<= t_0 4e+278)))
      (sqrt (/ (/ A (/ l c0_m)) (/ V c0_m)))
      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 <= 0.0) || !(t_0 <= 4e+278)) {
		tmp = sqrt(((A / (l / c0_m)) / (V / c0_m)));
	} 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 <= 0.0d0) .or. (.not. (t_0 <= 4d+278))) then
        tmp = sqrt(((a / (l / c0_m)) / (v / c0_m)))
    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 <= 0.0) || !(t_0 <= 4e+278)) {
		tmp = Math.sqrt(((A / (l / c0_m)) / (V / c0_m)));
	} 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 <= 0.0) or not (t_0 <= 4e+278):
		tmp = math.sqrt(((A / (l / c0_m)) / (V / c0_m)))
	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 <= 0.0) || !(t_0 <= 4e+278))
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) / Float64(V / c0_m)));
	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 <= 0.0) || ~((t_0 <= 4e+278)))
		tmp = sqrt(((A / (l / c0_m)) / (V / c0_m)));
	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, 0.0], N[Not[LessEqual[t$95$0, 4e+278]], $MachinePrecision]], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $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)))) < 0.0 or 3.99999999999999985e278 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.0%

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

      1. associate-/r* 68.0%

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

      2. div-inv 68.0%

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

      3. add-cube-cbrt 67.7%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 68.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 68.5%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 68.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 69.2%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 68.7%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 68.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 68.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 68.7%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 68.7%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 19.6%

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

      1. associate-*l/ 20.7%

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

      2. associate-/l* 20.8%

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

   10. Simplified 20.8%

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

       1. unpow2 20.8%

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

       2. *-commutative 20.8%

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

       3. times-frac 28.1%

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

   12. Applied egg-rr 28.1%

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

       1. *-commutative 28.1%

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

       2. clear-num 28.1%

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

       3. frac-times 27.6%

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

       4. *-un-lft-identity 27.6%

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

   14. Applied egg-rr 27.6%

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

       1. associate-*r/ 27.5%

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

       2. *-commutative 27.5%

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

       3. associate-/r* 28.5%

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

       4. clear-num 28.5%

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

       5. associate-/l/ 29.0%

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

       6. clear-num 29.0%

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

   16. Applied egg-rr 29.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e278

   1. Initial program 99.1%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+278}\right):\\ \;\;\;\;\sqrt{\frac{\frac{A}{\frac{\ell}{c0}}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{V} \cdot \frac{c0\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0) (not (<= t_0 5e+303)))
      (sqrt (* A (* (/ c0_m V) (/ c0_m 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 <= 0.0) || !(t_0 <= 5e+303)) {
		tmp = sqrt((A * ((c0_m / V) * (c0_m / 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 <= 0.0d0) .or. (.not. (t_0 <= 5d+303))) then
        tmp = sqrt((a * ((c0_m / v) * (c0_m / 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 <= 0.0) || !(t_0 <= 5e+303)) {
		tmp = Math.sqrt((A * ((c0_m / V) * (c0_m / 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 <= 0.0) or not (t_0 <= 5e+303):
		tmp = math.sqrt((A * ((c0_m / V) * (c0_m / 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 <= 0.0) || !(t_0 <= 5e+303))
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / V) * Float64(c0_m / 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 <= 0.0) || ~((t_0 <= 5e+303)))
		tmp = sqrt((A * ((c0_m / V) * (c0_m / 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, 0.0], N[Not[LessEqual[t$95$0, 5e+303]], $MachinePrecision]], N[Sqrt[N[(A * N[(N[(c0$95$m / V), $MachinePrecision] * N[(c0$95$m / l), $MachinePrecision]), $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)))) < 0.0 or 4.9999999999999997e303 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.8%

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

      1. associate-/r* 68.3%

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

      2. div-inv 68.3%

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

      3. add-cube-cbrt 68.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 68.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 68.3%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 68.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 69.5%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 69.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 69.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 69.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 69.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 69.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 19.6%

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

      1. associate-*l/ 20.8%

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

      2. associate-/l* 20.8%

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

   10. Simplified 20.8%

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

       1. unpow2 20.8%

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

       2. *-commutative 20.8%

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

       3. times-frac 28.2%

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

   12. Applied egg-rr 28.2%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999997e303

   1. Initial program 99.1%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\left(A \cdot c0\_m\right) \cdot \frac{c0\_m}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+278}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot c0\_m}{\ell \cdot \frac{V}{c0\_m}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0)
      (sqrt (/ (* (* A c0_m) (/ c0_m V)) l))
      (if (<= t_0 4e+278) t_0 (sqrt (/ (* A c0_m) (* l (/ V c0_m)))))))))

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 <= 0.0) {
		tmp = sqrt((((A * c0_m) * (c0_m / V)) / l));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = sqrt(((A * c0_m) / (l * (V / c0_m))));
	}
	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 <= 0.0d0) then
        tmp = sqrt((((a * c0_m) * (c0_m / v)) / l))
    else if (t_0 <= 4d+278) then
        tmp = t_0
    else
        tmp = sqrt(((a * c0_m) / (l * (v / c0_m))))
    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 <= 0.0) {
		tmp = Math.sqrt((((A * c0_m) * (c0_m / V)) / l));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(((A * c0_m) / (l * (V / c0_m))));
	}
	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 <= 0.0:
		tmp = math.sqrt((((A * c0_m) * (c0_m / V)) / l))
	elif t_0 <= 4e+278:
		tmp = t_0
	else:
		tmp = math.sqrt(((A * c0_m) / (l * (V / c0_m))))
	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 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(A * c0_m) * Float64(c0_m / V)) / l));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = sqrt(Float64(Float64(A * c0_m) / Float64(l * Float64(V / c0_m))));
	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 <= 0.0)
		tmp = sqrt((((A * c0_m) * (c0_m / V)) / l));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = sqrt(((A * c0_m) / (l * (V / c0_m))));
	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, 0.0], N[Sqrt[N[(N[(N[(A * c0$95$m), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+278], t$95$0, N[Sqrt[N[(N[(A * c0$95$m), $MachinePrecision] / N[(l * N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

      1. associate-/r* 71.5%

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

      2. div-inv 71.6%

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

      3. add-cube-cbrt 71.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 71.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 71.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 72.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 17.2%

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

      1. associate-*l/ 18.4%

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

      2. associate-/l* 18.5%

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

   10. Simplified 18.5%

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

       1. unpow2 18.5%

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

       2. associate-/l* 21.0%

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

   12. Applied egg-rr 21.0%

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

       1. associate-*r* 20.9%

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

       2. associate-/r* 22.6%

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

       3. associate-*r/ 23.8%

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

   14. Applied egg-rr 23.8%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e278

   1. Initial program 99.1%

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

   if 3.99999999999999985e278 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 41.7%

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

      1. associate-/r* 42.1%

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

      2. div-inv 42.1%

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

      3. add-cube-cbrt 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 45.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 45.9%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 45.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 42.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 37.0%

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

      1. associate-*l/ 37.5%

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

      2. associate-/l* 37.5%

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

   10. Simplified 37.5%

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

       1. unpow2 37.5%

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

       2. associate-/l* 45.7%

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

   12. Applied egg-rr 45.7%

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

       1. associate-*r* 49.8%

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

       2. clear-num 49.8%

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

       3. un-div-inv 49.8%

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

   14. Applied egg-rr 49.8%

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

       1. associate-/l* 59.0%

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

   16. Applied egg-rr 59.0%

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

       1. *-commutative 59.0%

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

       2. associate-*l/ 49.8%

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

       3. associate-*r/ 63.2%

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

   18. Simplified 63.2%

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

Alternative 6: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{V} \cdot \frac{c0\_m}{\ell}\right)}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+278}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot c0\_m}{\ell \cdot \frac{V}{c0\_m}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0)
      (sqrt (* A (* (/ c0_m V) (/ c0_m l))))
      (if (<= t_0 4e+278) t_0 (sqrt (/ (* A c0_m) (* l (/ V c0_m)))))))))

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 <= 0.0) {
		tmp = sqrt((A * ((c0_m / V) * (c0_m / l))));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = sqrt(((A * c0_m) / (l * (V / c0_m))));
	}
	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 <= 0.0d0) then
        tmp = sqrt((a * ((c0_m / v) * (c0_m / l))))
    else if (t_0 <= 4d+278) then
        tmp = t_0
    else
        tmp = sqrt(((a * c0_m) / (l * (v / c0_m))))
    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 <= 0.0) {
		tmp = Math.sqrt((A * ((c0_m / V) * (c0_m / l))));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(((A * c0_m) / (l * (V / c0_m))));
	}
	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 <= 0.0:
		tmp = math.sqrt((A * ((c0_m / V) * (c0_m / l))))
	elif t_0 <= 4e+278:
		tmp = t_0
	else:
		tmp = math.sqrt(((A * c0_m) / (l * (V / c0_m))))
	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 <= 0.0)
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / V) * Float64(c0_m / l))));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = sqrt(Float64(Float64(A * c0_m) / Float64(l * Float64(V / c0_m))));
	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 <= 0.0)
		tmp = sqrt((A * ((c0_m / V) * (c0_m / l))));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = sqrt(((A * c0_m) / (l * (V / c0_m))));
	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, 0.0], N[Sqrt[N[(A * N[(N[(c0$95$m / V), $MachinePrecision] * N[(c0$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+278], t$95$0, N[Sqrt[N[(N[(A * c0$95$m), $MachinePrecision] / N[(l * N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

      1. associate-/r* 71.5%

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

      2. div-inv 71.6%

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

      3. add-cube-cbrt 71.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 71.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 71.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 72.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 17.2%

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

      1. associate-*l/ 18.4%

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

      2. associate-/l* 18.5%

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

   10. Simplified 18.5%

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

       1. unpow2 18.5%

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

       2. *-commutative 18.5%

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

       3. times-frac 23.3%

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

   12. Applied egg-rr 23.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e278

   1. Initial program 99.1%

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

   if 3.99999999999999985e278 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 41.7%

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

      1. associate-/r* 42.1%

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

      2. div-inv 42.1%

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

      3. add-cube-cbrt 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 45.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 45.9%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 45.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 42.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 37.0%

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

      1. associate-*l/ 37.5%

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

      2. associate-/l* 37.5%

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

   10. Simplified 37.5%

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

       1. unpow2 37.5%

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

       2. associate-/l* 45.7%

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

   12. Applied egg-rr 45.7%

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

       1. associate-*r* 49.8%

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

       2. clear-num 49.8%

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

       3. un-div-inv 49.8%

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

   14. Applied egg-rr 49.8%

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

       1. associate-/l* 59.0%

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

   16. Applied egg-rr 59.0%

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

       1. *-commutative 59.0%

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

       2. associate-*l/ 49.8%

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

       3. associate-*r/ 63.2%

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

   18. Simplified 63.2%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+278}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A \cdot c0}{\ell \cdot \frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{V} \cdot \frac{c0\_m}{\ell}\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{c0\_m}{\ell \cdot \frac{V}{c0\_m}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0)
      (sqrt (* A (* (/ c0_m V) (/ c0_m l))))
      (if (<= t_0 2e+284) t_0 (sqrt (* A (/ c0_m (* l (/ V c0_m))))))))))

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 <= 0.0) {
		tmp = sqrt((A * ((c0_m / V) * (c0_m / l))));
	} else if (t_0 <= 2e+284) {
		tmp = t_0;
	} else {
		tmp = sqrt((A * (c0_m / (l * (V / c0_m)))));
	}
	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 <= 0.0d0) then
        tmp = sqrt((a * ((c0_m / v) * (c0_m / l))))
    else if (t_0 <= 2d+284) then
        tmp = t_0
    else
        tmp = sqrt((a * (c0_m / (l * (v / c0_m)))))
    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 <= 0.0) {
		tmp = Math.sqrt((A * ((c0_m / V) * (c0_m / l))));
	} else if (t_0 <= 2e+284) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((A * (c0_m / (l * (V / c0_m)))));
	}
	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 <= 0.0:
		tmp = math.sqrt((A * ((c0_m / V) * (c0_m / l))))
	elif t_0 <= 2e+284:
		tmp = t_0
	else:
		tmp = math.sqrt((A * (c0_m / (l * (V / c0_m)))))
	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 <= 0.0)
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / V) * Float64(c0_m / l))));
	elseif (t_0 <= 2e+284)
		tmp = t_0;
	else
		tmp = sqrt(Float64(A * Float64(c0_m / Float64(l * Float64(V / c0_m)))));
	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 <= 0.0)
		tmp = sqrt((A * ((c0_m / V) * (c0_m / l))));
	elseif (t_0 <= 2e+284)
		tmp = t_0;
	else
		tmp = sqrt((A * (c0_m / (l * (V / c0_m)))));
	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, 0.0], N[Sqrt[N[(A * N[(N[(c0$95$m / V), $MachinePrecision] * N[(c0$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2e+284], t$95$0, N[Sqrt[N[(A * N[(c0$95$m / N[(l * N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

      1. associate-/r* 71.5%

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

      2. div-inv 71.6%

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

      3. add-cube-cbrt 71.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 71.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 71.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 72.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 17.2%

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

      1. associate-*l/ 18.4%

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

      2. associate-/l* 18.5%

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

   10. Simplified 18.5%

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

       1. unpow2 18.5%

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

       2. *-commutative 18.5%

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

       3. times-frac 23.3%

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

   12. Applied egg-rr 23.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000016e284

   1. Initial program 99.1%

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

   if 2.00000000000000016e284 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 41.7%

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

      1. associate-/r* 42.1%

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

      2. div-inv 42.1%

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

      3. add-cube-cbrt 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 45.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 45.9%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 45.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 42.1%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 42.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 37.0%

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

      1. associate-*l/ 37.5%

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

      2. associate-/l* 37.5%

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

   10. Simplified 37.5%

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

       1. unpow2 37.5%

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

       2. *-commutative 37.5%

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

       3. times-frac 63.3%

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

   12. Applied egg-rr 63.3%

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

       1. *-commutative 63.3%

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

       2. clear-num 63.3%

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

       3. frac-times 63.1%

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

       4. *-un-lft-identity 63.1%

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

   14. Applied egg-rr 63.1%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+284}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{c0}{\ell \cdot \frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2.5 \cdot 10^{+212}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0) (not (<= t_0 2.5e+212)))
      (* 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 <= 0.0) || !(t_0 <= 2.5e+212)) {
		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 <= 0.0d0) .or. (.not. (t_0 <= 2.5d+212))) 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 <= 0.0) || !(t_0 <= 2.5e+212)) {
		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 <= 0.0) or not (t_0 <= 2.5e+212):
		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 <= 0.0) || !(t_0 <= 2.5e+212))
		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 <= 0.0) || ~((t_0 <= 2.5e+212)))
		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, 0.0], N[Not[LessEqual[t$95$0, 2.5e+212]], $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)))) < 0.0 or 2.49999999999999996e212 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.7%

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

      1. associate-/r* 68.7%

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

   3. Simplified 68.7%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.49999999999999996e212

   1. Initial program 99.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2.5 \cdot 10^{+212}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(c0\_m \cdot \frac{c0\_m}{V \cdot \ell}\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0)
      (sqrt (* A (* c0_m (/ c0_m (* V l)))))
      (if (<= t_0 1e+185) t_0 (/ c0_m (sqrt (* V (/ l 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 <= 0.0) {
		tmp = sqrt((A * (c0_m * (c0_m / (V * l)))));
	} else if (t_0 <= 1e+185) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((V * (l / 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 <= 0.0d0) then
        tmp = sqrt((a * (c0_m * (c0_m / (v * l)))))
    else if (t_0 <= 1d+185) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((v * (l / 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 <= 0.0) {
		tmp = Math.sqrt((A * (c0_m * (c0_m / (V * l)))));
	} else if (t_0 <= 1e+185) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((V * (l / 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 <= 0.0:
		tmp = math.sqrt((A * (c0_m * (c0_m / (V * l)))))
	elif t_0 <= 1e+185:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((V * (l / 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 <= 0.0)
		tmp = sqrt(Float64(A * Float64(c0_m * Float64(c0_m / Float64(V * l)))));
	elseif (t_0 <= 1e+185)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / 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 <= 0.0)
		tmp = sqrt((A * (c0_m * (c0_m / (V * l)))));
	elseif (t_0 <= 1e+185)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((V * (l / 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, 0.0], N[Sqrt[N[(A * N[(c0$95$m * N[(c0$95$m / N[(V * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+185], t$95$0, N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

      1. associate-/r* 71.5%

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

      2. div-inv 71.6%

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

      3. add-cube-cbrt 71.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 71.6%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 71.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 71.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 72.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 72.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 17.2%

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

      1. associate-*l/ 18.4%

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

      2. associate-/l* 18.5%

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

   10. Simplified 18.5%

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

       1. unpow2 18.5%

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

       2. associate-/l* 21.0%

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

   12. Applied egg-rr 21.0%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999998e184

   1. Initial program 99.1%

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

   if 9.9999999999999998e184 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.6%

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

      1. *-un-lft-identity 56.6%

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

      2. times-frac 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.7%

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

      2. *-un-lft-identity 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. div-inv 59.7%

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

      2. clear-num 59.7%

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

      3. frac-times 59.7%

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

      4. *-commutative 59.7%

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

      5. add-sqr-sqrt 59.6%

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

      6. frac-times 59.7%

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

      7. sqrt-unprod 60.5%

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

      8. add-sqr-sqrt 60.6%

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

      9. un-div-inv 60.6%

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

   8. Applied egg-rr 60.6%

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

Alternative 10: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 0.0)
      (* c0_m (sqrt (/ (/ A V) l)))
      (if (<= t_0 1e+185) t_0 (/ c0_m (sqrt (* V (/ l 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 <= 0.0) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+185) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((V * (l / 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 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / v) / l))
    else if (t_0 <= 1d+185) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((v * (l / 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 <= 0.0) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+185) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((V * (l / 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 <= 0.0:
		tmp = c0_m * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+185:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((V * (l / 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 <= 0.0)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+185)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / 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 <= 0.0)
		tmp = c0_m * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+185)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((V * (l / 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, 0.0], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+185], t$95$0, N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999998e184

   1. Initial program 99.1%

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

   if 9.9999999999999998e184 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.6%

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

      1. *-un-lft-identity 56.6%

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

      2. times-frac 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.7%

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

      2. *-un-lft-identity 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. div-inv 59.7%

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

      2. clear-num 59.7%

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

      3. frac-times 59.7%

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

      4. *-commutative 59.7%

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

      5. add-sqr-sqrt 59.6%

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

      6. frac-times 59.7%

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

      7. sqrt-unprod 60.5%

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

      8. add-sqr-sqrt 60.6%

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

      9. un-div-inv 60.6%

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

   8. Applied egg-rr 60.6%

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

Alternative 11: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \leq -9 \cdot 10^{-297}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 -9e-297)
    (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
    (* c0_m (/ (sqrt A) (* (sqrt l) (sqrt 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 tmp;
	if (V <= -9e-297) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else {
		tmp = c0_m * (sqrt(A) / (sqrt(l) * sqrt(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) :: tmp
    if (v <= (-9d-297)) then
        tmp = c0_m * (sqrt((a / -l)) / sqrt(-v))
    else
        tmp = c0_m * (sqrt(a) / (sqrt(l) * sqrt(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 tmp;
	if (V <= -9e-297) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else {
		tmp = c0_m * (Math.sqrt(A) / (Math.sqrt(l) * Math.sqrt(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):
	tmp = 0
	if V <= -9e-297:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	else:
		tmp = c0_m * (math.sqrt(A) / (math.sqrt(l) * math.sqrt(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)
	tmp = 0.0
	if (V <= -9e-297)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / Float64(sqrt(l) * sqrt(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)
	tmp = 0.0;
	if (V <= -9e-297)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = c0_m * (sqrt(A) / (sqrt(l) * sqrt(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_] := N[(c0$95$s * If[LessEqual[V, -9e-297], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if V < -8.99999999999999951e-297

   1. Initial program 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. times-frac 77.1%

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

   4. Applied egg-rr 77.1%

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

      1. associate-*l/ 77.1%

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

      2. *-un-lft-identity 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. frac-2neg 77.1%

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

      2. sqrt-div 91.5%

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

   8. Applied egg-rr 91.5%

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

      1. distribute-frac-neg2 91.5%

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

   10. Simplified 91.5%

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

   if -8.99999999999999951e-297 < V

   1. Initial program 72.5%

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

      1. associate-/r* 73.4%

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

      2. div-inv 73.4%

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

      3. add-cube-cbrt 73.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 74.8%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 74.8%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 74.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 76.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 75.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 75.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 75.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 75.4%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 75.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 73.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      2. associate-/l/ 73.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}}} \]

      3. unpow2 73.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot \sqrt[3]{\ell}}} \]

      4. add-cube-cbrt 73.4%

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

      5. un-div-inv 73.4%

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

      6. sqrt-unprod 45.5%

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

      7. *-commutative 45.5%

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

      8. sqrt-div 56.0%

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

      9. sqrt-div 55.9%

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

      10. metadata-eval 55.9%

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

      11. frac-times 56.0%

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

      12. *-un-lft-identity 56.0%

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

   8. Applied egg-rr 56.0%

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

Alternative 12: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{A \cdot \frac{c0\_m \cdot \frac{c0\_m}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-300}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{1}{\sqrt{\frac{V}{c0\_m} \cdot \frac{\frac{\ell}{c0\_m}}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+209}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\frac{\ell}{c0\_m}}}{\frac{V}{c0\_m}}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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))
    (sqrt (* A (/ (* c0_m (/ c0_m V)) l)))
    (if (<= (* V l) -5e-300)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 0.0)
        (/ 1.0 (sqrt (* (/ V c0_m) (/ (/ l c0_m) A))))
        (if (<= (* V l) 5e+209)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (/ (/ A (/ l c0_m)) (/ V c0_m)))))))))

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 = sqrt((A * ((c0_m * (c0_m / V)) / l)));
	} else if ((V * l) <= -5e-300) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = 1.0 / sqrt(((V / c0_m) * ((l / c0_m) / A)));
	} else if ((V * l) <= 5e+209) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((A / (l / c0_m)) / (V / c0_m)));
	}
	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 = Math.sqrt((A * ((c0_m * (c0_m / V)) / l)));
	} else if ((V * l) <= -5e-300) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = 1.0 / Math.sqrt(((V / c0_m) * ((l / c0_m) / A)));
	} else if ((V * l) <= 5e+209) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((A / (l / c0_m)) / (V / c0_m)));
	}
	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 = math.sqrt((A * ((c0_m * (c0_m / V)) / l)))
	elif (V * l) <= -5e-300:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = 1.0 / math.sqrt(((V / c0_m) * ((l / c0_m) / A)))
	elif (V * l) <= 5e+209:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((A / (l / c0_m)) / (V / c0_m)))
	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 = sqrt(Float64(A * Float64(Float64(c0_m * Float64(c0_m / V)) / l)));
	elseif (Float64(V * l) <= -5e-300)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(1.0 / sqrt(Float64(Float64(V / c0_m) * Float64(Float64(l / c0_m) / A))));
	elseif (Float64(V * l) <= 5e+209)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64(A / Float64(l / c0_m)) / Float64(V / c0_m)));
	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 = sqrt((A * ((c0_m * (c0_m / V)) / l)));
	elseif ((V * l) <= -5e-300)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = 1.0 / sqrt(((V / c0_m) * ((l / c0_m) / A)));
	elseif ((V * l) <= 5e+209)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(((A / (l / c0_m)) / (V / c0_m)));
	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[Sqrt[N[(A * N[(N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-300], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(1.0 / N[Sqrt[N[(N[(V / c0$95$m), $MachinePrecision] * N[(N[(l / c0$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+209], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(A / N[(l / c0$95$m), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 36.5%

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

      1. associate-/r* 55.7%

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

      2. div-inv 55.7%

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

      3. add-cube-cbrt 55.6%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 49.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 49.3%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 49.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 49.3%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 49.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 49.3%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 49.3%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 49.3%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 49.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 35.0%

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

      1. associate-*l/ 34.8%

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

      2. associate-/l* 35.0%

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

   10. Simplified 35.0%

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

       1. unpow2 35.0%

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

       2. associate-/l* 36.5%

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

   12. Applied egg-rr 36.5%

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

       1. *-commutative 36.5%

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

       2. associate-/r* 42.3%

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

       3. associate-*l/ 48.8%

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

   14. Applied egg-rr 48.8%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999996e-300

   1. Initial program 92.5%

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

      1. frac-2neg 92.5%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if -4.99999999999999996e-300 < (*.f64 V l) < 0.0

   1. Initial program 32.2%

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

      1. associate-/r* 52.2%

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

      2. div-inv 52.2%

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

      3. add-cube-cbrt 52.1%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 49.0%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 49.0%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 49.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 48.9%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 48.9%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 49.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 49.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 49.0%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 49.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 6.2%

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

      1. associate-*l/ 4.2%

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

      2. associate-/l* 6.3%

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

   10. Simplified 6.3%

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

       1. unpow2 6.3%

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

       2. associate-/l* 9.5%

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

   12. Applied egg-rr 9.5%

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

       1. associate-*r* 9.1%

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

       2. clear-num 9.1%

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

       3. un-div-inv 9.1%

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

   14. Applied egg-rr 9.1%

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

       1. clear-num 9.1%

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

       2. sqrt-div 9.1%

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

       3. metadata-eval 9.1%

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

       4. associate-/l* 22.6%

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

       5. *-commutative 22.6%

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

       6. times-frac 30.1%

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

   16. Applied egg-rr 30.1%

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

   if 0.0 < (*.f64 V l) < 4.99999999999999964e209

   1. Initial program 83.2%

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

      1. sqrt-div 98.9%

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

      2. div-inv 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   6. Simplified 98.9%

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

   if 4.99999999999999964e209 < (*.f64 V l)

   1. Initial program 56.3%

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

      1. associate-/r* 69.3%

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

      2. div-inv 69.3%

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

      3. add-cube-cbrt 69.2%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \]

      4. times-frac 65.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]

      5. pow2 65.4%

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}} \]

   4. Applied egg-rr 65.4%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 65.4%

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot \frac{\frac{1}{V}}{\sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

      2. associate-/l/ 62.3%

         \[\leadsto c0 \cdot \sqrt{\frac{A \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      3. associate-*r/ 62.3%

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A \cdot 1}{\sqrt[3]{\ell} \cdot V}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      4. *-rgt-identity 62.3%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{A}}{\sqrt[3]{\ell} \cdot V}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

      5. *-commutative 62.3%

         \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{V \cdot \sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]

   6. Simplified 62.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   8. Applied egg-rr 34.0%

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

      1. associate-*l/ 34.1%

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

      2. associate-/l* 34.2%

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

   10. Simplified 34.2%

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

       1. unpow2 34.2%

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

       2. *-commutative 34.2%

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

       3. times-frac 53.3%

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

   12. Applied egg-rr 53.3%

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

       1. *-commutative 53.3%

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

       2. clear-num 53.2%

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

       3. frac-times 53.3%

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

       4. *-un-lft-identity 53.3%

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

   14. Applied egg-rr 53.3%

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

       1. associate-*r/ 53.2%

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

       2. *-commutative 53.2%

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

       3. associate-/r* 59.4%

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

       4. clear-num 59.4%

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

       5. associate-/l/ 59.5%

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

       6. clear-num 59.5%

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

   16. Applied egg-rr 59.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{A \cdot \frac{c0 \cdot \frac{c0}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{1}{\sqrt{\frac{V}{c0} \cdot \frac{\frac{\ell}{c0}}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+209}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\frac{\ell}{c0}}}{\frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \leq 10^{-288}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \leq 9 \cdot 10^{-214}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V} \cdot \sqrt{\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) 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 -4e-310)
    (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
    (if (<= V 1e-288)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= V 9e-214)
        (/ c0_m (* (sqrt V) (sqrt (/ l A))))
        (/ c0_m (* (sqrt l) (sqrt (/ 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 tmp;
	if (V <= -4e-310) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if (V <= 1e-288) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if (V <= 9e-214) {
		tmp = c0_m / (sqrt(V) * sqrt((l / A)));
	} else {
		tmp = c0_m / (sqrt(l) * sqrt((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) :: tmp
    if (v <= (-4d-310)) then
        tmp = c0_m * (sqrt((a / -l)) / sqrt(-v))
    else if (v <= 1d-288) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if (v <= 9d-214) then
        tmp = c0_m / (sqrt(v) * sqrt((l / a)))
    else
        tmp = c0_m / (sqrt(l) * sqrt((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 tmp;
	if (V <= -4e-310) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if (V <= 1e-288) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if (V <= 9e-214) {
		tmp = c0_m / (Math.sqrt(V) * Math.sqrt((l / A)));
	} else {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((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):
	tmp = 0
	if V <= -4e-310:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif V <= 1e-288:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif V <= 9e-214:
		tmp = c0_m / (math.sqrt(V) * math.sqrt((l / A)))
	else:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((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)
	tmp = 0.0
	if (V <= -4e-310)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (V <= 1e-288)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (V <= 9e-214)
		tmp = Float64(c0_m / Float64(sqrt(V) * sqrt(Float64(l / A))));
	else
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(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)
	tmp = 0.0;
	if (V <= -4e-310)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif (V <= 1e-288)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif (V <= 9e-214)
		tmp = c0_m / (sqrt(V) * sqrt((l / A)));
	else
		tmp = c0_m / (sqrt(l) * sqrt((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_] := N[(c0$95$s * If[LessEqual[V, -4e-310], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[V, 1e-288], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[V, 9e-214], N[(c0$95$m / N[(N[Sqrt[V], $MachinePrecision] * N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if V < -3.999999999999988e-310

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 76.9%

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

   4. Applied egg-rr 76.9%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. frac-2neg 76.9%

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

      2. sqrt-div 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. distribute-frac-neg2 91.7%

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

   10. Simplified 91.7%

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

   if -3.999999999999988e-310 < V < 1.00000000000000006e-288

   1. Initial program 100.0%

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

   if 1.00000000000000006e-288 < V < 9.0000000000000002e-214

   1. Initial program 52.9%

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

      1. *-un-lft-identity 52.9%

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

      2. times-frac 61.8%

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

   4. Applied egg-rr 61.8%

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

      1. associate-*l/ 61.7%

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

      2. *-un-lft-identity 61.7%

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

   6. Applied egg-rr 61.7%

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

      1. div-inv 61.8%

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

      2. clear-num 61.8%

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

      3. frac-times 61.7%

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

      4. *-commutative 61.7%

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

      5. add-sqr-sqrt 61.7%

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

      6. frac-times 61.7%

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

      7. sqrt-unprod 61.2%

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

      8. add-sqr-sqrt 61.7%

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

      9. un-div-inv 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. associate-*r/ 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-/l* 61.9%

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

   10. Simplified 61.9%

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

       1. associate-*r/ 52.9%

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

       2. *-commutative 52.9%

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

       3. associate-*r/ 61.8%

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

       4. sqrt-prod 89.9%

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

   12. Applied egg-rr 89.9%

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

       1. *-commutative 89.9%

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

   14. Simplified 89.9%

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

   if 9.0000000000000002e-214 < V

   1. Initial program 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. times-frac 74.7%

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

   4. Applied egg-rr 74.7%

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

      1. associate-*l/ 74.7%

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

      2. *-un-lft-identity 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. div-inv 74.7%

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

      2. clear-num 74.4%

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

      3. frac-times 74.4%

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

      4. *-commutative 74.4%

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

      5. add-sqr-sqrt 74.3%

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

      6. frac-times 74.4%

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

      7. sqrt-unprod 74.4%

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

      8. add-sqr-sqrt 74.6%

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

      9. un-div-inv 74.6%

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

   8. Applied egg-rr 74.6%

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

      1. associate-*r/ 74.7%

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

      2. *-commutative 74.7%

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

      3. associate-/l* 75.5%

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

   10. Simplified 75.5%

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

       1. *-commutative 75.5%

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

       2. sqrt-prod 45.2%

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

   12. Applied egg-rr 45.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \leq 10^{-288}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \leq 9 \cdot 10^{-214}:\\ \;\;\;\;\frac{c0}{\sqrt{V} \cdot \sqrt{\frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.7% 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 #s(literal 1 binary64) 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 74.4%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))