Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 91.8%

Time: 12.9s

Alternatives: 15

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: 73.5% 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: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{A}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+293}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t\_0}{\ell} \cdot \frac{t\_0}{V}}\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    c0_s
    (if (<= (* V l) (- INFINITY))
      (* (/ c0_m (sqrt l)) (sqrt (/ A V)))
      (if (<= (* V l) -1e-295)
        (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
        (if (<= (* V l) 5e-319)
          (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
          (if (<= (* V l) 5e+293)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (sqrt (* (/ t_0 l) (/ t_0 V))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt(A);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -1e-295) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= 5e+293) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((t_0 / l) * (t_0 / V)));
	}
	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 t_0 = c0_m * Math.sqrt(A);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -1e-295) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= 5e+293) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((t_0 / l) * (t_0 / V)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt(A)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -1e-295:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-319:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= 5e+293:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((t_0 / l) * (t_0 / V)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -1e-295)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-319)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 5e+293)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64(t_0 / l) * Float64(t_0 / V)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt(A);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -1e-295)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-319)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= 5e+293)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(((t_0 / l) * (t_0 / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-295], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-319], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+293], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000006e-295

   1. Initial program 89.0%

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

      1. frac-2neg 89.0%

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

      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 -1.00000000000000006e-295 < (*.f64 V l) < 4.9999937e-319

   1. Initial program 32.7%

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

      1. clear-num 32.7%

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

      2. associate-/r/ 30.9%

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

      3. associate-/r* 30.9%

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

   4. Applied egg-rr 30.9%

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

      1. associate-*l/ 59.5%

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

      2. associate-*r/ 59.4%

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

      3. associate-*l/ 59.5%

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

      4. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. frac-2neg 59.5%

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

      2. sqrt-div 42.5%

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

      3. distribute-neg-frac2 42.5%

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

   8. Applied egg-rr 42.5%

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

   if 4.9999937e-319 < (*.f64 V l) < 5.00000000000000033e293

   1. Initial program 87.5%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 5.00000000000000033e293 < (*.f64 V l)

   1. Initial program 20.2%

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

      1. add-sqr-sqrt 20.2%

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

      2. sqrt-unprod 20.2%

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

      3. *-commutative 20.2%

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

      4. *-commutative 20.2%

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

      5. swap-sqr 19.2%

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

      6. add-sqr-sqrt 19.2%

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

      7. pow2 19.2%

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

   4. Applied egg-rr 19.2%

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

      1. associate-/r* 19.1%

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

   6. Simplified 19.1%

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

      1. *-commutative 19.1%

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

      2. associate-/l/ 19.2%

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

      3. associate-*r/ 18.9%

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

      4. *-commutative 18.9%

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

   8. Applied egg-rr 18.9%

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

      1. add-sqr-sqrt 18.9%

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

      2. *-commutative 18.9%

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

      3. times-frac 25.9%

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

      4. sqrt-prod 25.9%

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

      5. sqrt-pow1 25.5%

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

      6. metadata-eval 25.5%

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

      7. pow1 25.5%

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

      8. sqrt-prod 31.7%

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

      9. sqrt-pow1 45.3%

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

      10. metadata-eval 45.3%

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

      11. pow1 45.3%

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

   10. Applied egg-rr 45.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \sqrt{A}}{\ell} \cdot \frac{c0 \cdot \sqrt{A}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+234}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \end{array} \]

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 l) (- INFINITY))
    (* (/ c0_m (sqrt l)) (sqrt (/ A V)))
    (if (<= (* V l) -1e-295)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 5e-319)
        (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
        (if (<= (* V l) 2e+234)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (* 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 * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -1e-295) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= 2e+234) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m * (sqrt(A) / (sqrt(l) * sqrt(V)));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -1e-295) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= 2e+234) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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 * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -1e-295:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-319:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= 2e+234:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	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 (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -1e-295)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-319)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 2e+234)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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 * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -1e-295)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-319)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= 2e+234)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	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[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-295], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-319], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+234], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $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 5 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000006e-295

   1. Initial program 89.0%

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

      1. frac-2neg 89.0%

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

      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 -1.00000000000000006e-295 < (*.f64 V l) < 4.9999937e-319

   1. Initial program 32.7%

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

      1. clear-num 32.7%

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

      2. associate-/r/ 30.9%

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

      3. associate-/r* 30.9%

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

   4. Applied egg-rr 30.9%

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

      1. associate-*l/ 59.5%

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

      2. associate-*r/ 59.4%

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

      3. associate-*l/ 59.5%

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

      4. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. frac-2neg 59.5%

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

      2. sqrt-div 42.5%

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

      3. distribute-neg-frac2 42.5%

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

   8. Applied egg-rr 42.5%

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

   if 4.9999937e-319 < (*.f64 V l) < 2.00000000000000004e234

   1. Initial program 88.3%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 2.00000000000000004e234 < (*.f64 V l)

   1. Initial program 23.8%

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

      1. clear-num 23.8%

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

      2. associate-/r/ 23.8%

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

      3. associate-/r* 27.1%

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

   4. Applied egg-rr 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-/l/ 23.8%

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

      3. div-inv 23.8%

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

      4. frac-2neg 23.8%

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

      5. distribute-rgt-neg-out 23.8%

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

      6. sqrt-undiv 0.0%

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

      7. associate-*r/ 0.0%

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

      8. sqrt-prod 0.0%

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

      9. associate-/r* 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. sqr-neg 0.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 19.8%

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

      17. sqr-neg 19.8%

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

      18. sqrt-unprod 58.3%

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

      19. add-sqr-sqrt 58.3%

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

   6. Applied egg-rr 58.3%

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

      1. associate-/l/ 58.5%

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

      2. associate-/l* 58.6%

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

      3. *-commutative 58.6%

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

   8. Simplified 58.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+234}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-V}\\ t_1 := c0\_m \cdot \sqrt{A}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0\_m \cdot \frac{\frac{1}{\frac{t\_0}{\sqrt{-A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{t\_0}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+293}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t\_1}{\ell} \cdot \frac{t\_1}{V}}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (- V))) (t_1 (* c0_m (sqrt A))))
   (*
    c0_s
    (if (<= (* V l) -1e-295)
      (* c0_m (/ (/ 1.0 (/ t_0 (sqrt (- A)))) (sqrt l)))
      (if (<= (* V l) 5e-319)
        (* c0_m (/ (sqrt (/ A (- l))) t_0))
        (if (<= (* V l) 5e+293)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (* (/ t_1 l) (/ t_1 V)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = sqrt(-V);
	double t_1 = c0_m * sqrt(A);
	double tmp;
	if ((V * l) <= -1e-295) {
		tmp = c0_m * ((1.0 / (t_0 / sqrt(-A))) / sqrt(l));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (sqrt((A / -l)) / t_0);
	} else if ((V * l) <= 5e+293) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((t_1 / l) * (t_1 / V)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-v)
    t_1 = c0_m * sqrt(a)
    if ((v * l) <= (-1d-295)) then
        tmp = c0_m * ((1.0d0 / (t_0 / sqrt(-a))) / sqrt(l))
    else if ((v * l) <= 5d-319) then
        tmp = c0_m * (sqrt((a / -l)) / t_0)
    else if ((v * l) <= 5d+293) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt(((t_1 / l) * (t_1 / v)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = Math.sqrt(-V);
	double t_1 = c0_m * Math.sqrt(A);
	double tmp;
	if ((V * l) <= -1e-295) {
		tmp = c0_m * ((1.0 / (t_0 / Math.sqrt(-A))) / Math.sqrt(l));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (Math.sqrt((A / -l)) / t_0);
	} else if ((V * l) <= 5e+293) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((t_1 / l) * (t_1 / V)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = math.sqrt(-V)
	t_1 = c0_m * math.sqrt(A)
	tmp = 0
	if (V * l) <= -1e-295:
		tmp = c0_m * ((1.0 / (t_0 / math.sqrt(-A))) / math.sqrt(l))
	elif (V * l) <= 5e-319:
		tmp = c0_m * (math.sqrt((A / -l)) / t_0)
	elif (V * l) <= 5e+293:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((t_1 / l) * (t_1 / V)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(Float64(-V))
	t_1 = Float64(c0_m * sqrt(A))
	tmp = 0.0
	if (Float64(V * l) <= -1e-295)
		tmp = Float64(c0_m * Float64(Float64(1.0 / Float64(t_0 / sqrt(Float64(-A)))) / sqrt(l)));
	elseif (Float64(V * l) <= 5e-319)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / t_0));
	elseif (Float64(V * l) <= 5e+293)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64(t_1 / l) * Float64(t_1 / V)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(-V);
	t_1 = c0_m * sqrt(A);
	tmp = 0.0;
	if ((V * l) <= -1e-295)
		tmp = c0_m * ((1.0 / (t_0 / sqrt(-A))) / sqrt(l));
	elseif ((V * l) <= 5e-319)
		tmp = c0_m * (sqrt((A / -l)) / t_0);
	elseif ((V * l) <= 5e+293)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(((t_1 / l) * (t_1 / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[(-V)], $MachinePrecision]}, Block[{t$95$1 = N[(c0$95$m * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e-295], N[(c0$95$m * N[(N[(1.0 / N[(t$95$0 / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-319], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+293], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$1 / l), $MachinePrecision] * N[(t$95$1 / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.00000000000000006e-295

   1. Initial program 81.0%

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

      1. associate-/r* 74.9%

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

      2. sqrt-div 35.1%

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

      3. div-inv 35.1%

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

   4. Applied egg-rr 35.1%

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

      1. associate-*r/ 35.1%

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

      2. *-rgt-identity 35.1%

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

   6. Simplified 35.1%

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

      1. clear-num 35.2%

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

      2. sqrt-div 35.2%

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

      3. metadata-eval 35.2%

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

   8. Applied egg-rr 35.2%

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

      1. frac-2neg 35.2%

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

      2. sqrt-div 41.2%

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

   10. Applied egg-rr 41.2%

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

   if -1.00000000000000006e-295 < (*.f64 V l) < 4.9999937e-319

   1. Initial program 32.7%

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

      1. clear-num 32.7%

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

      2. associate-/r/ 30.9%

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

      3. associate-/r* 30.9%

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

   4. Applied egg-rr 30.9%

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

      1. associate-*l/ 59.5%

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

      2. associate-*r/ 59.4%

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

      3. associate-*l/ 59.5%

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

      4. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. frac-2neg 59.5%

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

      2. sqrt-div 42.5%

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

      3. distribute-neg-frac2 42.5%

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

   8. Applied egg-rr 42.5%

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

   if 4.9999937e-319 < (*.f64 V l) < 5.00000000000000033e293

   1. Initial program 87.5%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 5.00000000000000033e293 < (*.f64 V l)

   1. Initial program 20.2%

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

      1. add-sqr-sqrt 20.2%

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

      2. sqrt-unprod 20.2%

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

      3. *-commutative 20.2%

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

      4. *-commutative 20.2%

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

      5. swap-sqr 19.2%

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

      6. add-sqr-sqrt 19.2%

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

      7. pow2 19.2%

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

   4. Applied egg-rr 19.2%

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

      1. associate-/r* 19.1%

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

   6. Simplified 19.1%

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

      1. *-commutative 19.1%

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

      2. associate-/l/ 19.2%

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

      3. associate-*r/ 18.9%

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

      4. *-commutative 18.9%

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

   8. Applied egg-rr 18.9%

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

      1. add-sqr-sqrt 18.9%

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

      2. *-commutative 18.9%

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

      3. times-frac 25.9%

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

      4. sqrt-prod 25.9%

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

      5. sqrt-pow1 25.5%

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

      6. metadata-eval 25.5%

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

      7. pow1 25.5%

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

      8. sqrt-prod 31.7%

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

      9. sqrt-pow1 45.3%

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

      10. metadata-eval 45.3%

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

      11. pow1 45.3%

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

   10. Applied egg-rr 45.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\frac{1}{\frac{\sqrt{-V}}{\sqrt{-A}}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \sqrt{A}}{\ell} \cdot \frac{c0 \cdot \sqrt{A}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 5e+274) 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 <= 5e+274) {
		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 <= 5d+274) 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 <= 5e+274) {
		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 <= 5e+274:
		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 <= 5e+274)
		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 <= 5e+274)
		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, 5e+274], 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 70.8%

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

      1. associate-/r* 72.4%

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

   3. Simplified 72.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999998e274

   1. Initial program 99.3%

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

   if 4.9999999999999998e274 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 45.2%

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

      1. clear-num 45.2%

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

      2. associate-/r/ 45.2%

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

      3. associate-/r* 45.2%

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

   4. Applied egg-rr 45.2%

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

      1. associate-*l/ 50.0%

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

      2. sqrt-div 37.8%

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

      3. associate-*l/ 37.7%

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

      4. *-un-lft-identity 37.7%

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

      5. clear-num 37.9%

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

      6. un-div-inv 37.7%

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

      7. sqrt-undiv 51.2%

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

      8. div-inv 51.1%

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

      9. clear-num 51.1%

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

   6. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. associate-*l/ 45.2%

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

      3. associate-*r/ 51.1%

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

   8. Simplified 51.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-305}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 5e-305)
      (* c0_m (sqrt (* (/ A V) (/ 1.0 l))))
      (if (<= t_0 5e+274) 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 <= 5e-305) {
		tmp = c0_m * sqrt(((A / V) * (1.0 / l)));
	} else if (t_0 <= 5e+274) {
		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 <= 5d-305) then
        tmp = c0_m * sqrt(((a / v) * (1.0d0 / l)))
    else if (t_0 <= 5d+274) 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 <= 5e-305) {
		tmp = c0_m * Math.sqrt(((A / V) * (1.0 / l)));
	} else if (t_0 <= 5e+274) {
		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 <= 5e-305:
		tmp = c0_m * math.sqrt(((A / V) * (1.0 / l)))
	elif t_0 <= 5e+274:
		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 <= 5e-305)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) * Float64(1.0 / l))));
	elseif (t_0 <= 5e+274)
		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 <= 5e-305)
		tmp = c0_m * sqrt(((A / V) * (1.0 / l)));
	elseif (t_0 <= 5e+274)
		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, 5e-305], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+274], 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)))) < 4.99999999999999985e-305

   1. Initial program 70.8%

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

      1. associate-/r* 72.4%

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

      2. div-inv 72.3%

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

   4. Applied egg-rr 72.3%

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

   if 4.99999999999999985e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999998e274

   1. Initial program 99.3%

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

   if 4.9999999999999998e274 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 45.2%

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

      1. clear-num 45.2%

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

      2. associate-/r/ 45.2%

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

      3. associate-/r* 45.2%

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

   4. Applied egg-rr 45.2%

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

      1. associate-*l/ 50.0%

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

      2. sqrt-div 37.8%

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

      3. associate-*l/ 37.7%

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

      4. *-un-lft-identity 37.7%

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

      5. clear-num 37.9%

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

      6. un-div-inv 37.7%

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

      7. sqrt-undiv 51.2%

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

      8. div-inv 51.1%

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

      9. clear-num 51.1%

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

   6. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. associate-*l/ 45.2%

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

      3. associate-*r/ 51.1%

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

   8. Simplified 51.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+255}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #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)) (sqrt l)))))
   (*
    c0_s
    (if (<= (* V l) -1e+202)
      t_0
      (if (<= (* V l) -1e-108)
        (* c0_m (sqrt (/ A (* V l))))
        (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+255)))
          t_0
          (* c0_m (/ (sqrt A) (sqrt (* V l))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -1e+202) {
		tmp = t_0;
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+255)) {
		tmp = t_0;
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-1d+202)) then
        tmp = t_0
    else if ((v * l) <= (-1d-108)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+255))) then
        tmp = t_0
    else
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -1e+202) {
		tmp = t_0;
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+255)) {
		tmp = t_0;
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -1e+202:
		tmp = t_0
	elif (V * l) <= -1e-108:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+255):
		tmp = t_0
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -1e+202)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-108)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+255))
		tmp = t_0;
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -1e+202)
		tmp = t_0;
	elseif ((V * l) <= -1e-108)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+255)))
		tmp = t_0;
	else
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+202], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-108], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+255]], $MachinePrecision]], t$95$0, N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.999999999999999e201 or -1.00000000000000004e-108 < (*.f64 V l) < -0.0 or 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 49.4%

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

      1. associate-/r* 63.2%

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

      2. sqrt-div 35.7%

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

      3. div-inv 35.6%

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

   4. Applied egg-rr 35.6%

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

      1. associate-*r/ 35.7%

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

      2. *-rgt-identity 35.7%

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

   6. Simplified 35.7%

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

   if -9.999999999999999e201 < (*.f64 V l) < -1.00000000000000004e-108

   1. Initial program 96.2%

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

   if -0.0 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 98.8%

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

   4. Applied egg-rr 98.8%

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

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+202}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+255}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+255}\right):\\ \;\;\;\;c0\_m \cdot \frac{t\_0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #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 (sqrt (/ A V))))
   (*
    c0_s
    (if (<= (* V l) (- INFINITY))
      (* (/ c0_m (sqrt l)) t_0)
      (if (<= (* V l) -1e-108)
        (* c0_m (sqrt (/ A (* V l))))
        (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+255)))
          (* c0_m (/ t_0 (sqrt l)))
          (* c0_m (/ (sqrt A) (sqrt (* V l))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * t_0;
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+255)) {
		tmp = c0_m * (t_0 / sqrt(l));
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * t_0;
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+255)) {
		tmp = c0_m * (t_0 / Math.sqrt(l));
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * t_0
	elif (V * l) <= -1e-108:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+255):
		tmp = c0_m * (t_0 / math.sqrt(l))
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * t_0);
	elseif (Float64(V * l) <= -1e-108)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+255))
		tmp = Float64(c0_m * Float64(t_0 / sqrt(l)));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * t_0;
	elseif ((V * l) <= -1e-108)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+255)))
		tmp = c0_m * (t_0 / sqrt(l));
	else
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-108], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+255]], $MachinePrecision]], N[(c0$95$m * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000004e-108

   1. Initial program 95.1%

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

   if -1.00000000000000004e-108 < (*.f64 V l) < -0.0 or 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 49.7%

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

      1. associate-/r* 63.8%

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

      2. sqrt-div 35.4%

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

      3. div-inv 35.4%

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

   4. Applied egg-rr 35.4%

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

      1. associate-*r/ 35.4%

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

      2. *-rgt-identity 35.4%

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

   6. Simplified 35.4%

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

   if -0.0 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 98.8%

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

   4. Applied egg-rr 98.8%

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

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+255}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0\_m \cdot \frac{t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (*
    c0_s
    (if (<= (* V l) (- INFINITY))
      (* (/ c0_m (sqrt l)) t_0)
      (if (<= (* V l) -1e-108)
        (* c0_m (sqrt (/ A (* V l))))
        (if (<= (* V l) 0.0)
          (* c0_m (/ t_0 (sqrt l)))
          (if (<= (* V l) 1e+255)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (/ (/ c0_m (sqrt V)) (sqrt (/ 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 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * t_0;
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m * (t_0 / sqrt(l));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0_m / sqrt(V)) / sqrt((l / A));
	}
	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 t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * t_0;
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0_m / Math.sqrt(V)) / Math.sqrt((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 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * t_0
	elif (V * l) <= -1e-108:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m * (t_0 / math.sqrt(l))
	elif (V * l) <= 1e+255:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0_m / math.sqrt(V)) / math.sqrt((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 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * t_0);
	elseif (Float64(V * l) <= -1e-108)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= 1e+255)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0_m / sqrt(V)) / sqrt(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 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * t_0;
	elseif ((V * l) <= -1e-108)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m * (t_0 / sqrt(l));
	elseif ((V * l) <= 1e+255)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0_m / sqrt(V)) / sqrt((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[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-108], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+255], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0$95$m / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000004e-108

   1. Initial program 95.1%

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

   if -1.00000000000000004e-108 < (*.f64 V l) < -0.0

   1. Initial program 55.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div 31.9%

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

      3. div-inv 31.9%

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

   4. Applied egg-rr 31.9%

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

      1. associate-*r/ 31.9%

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

      2. *-rgt-identity 31.9%

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

   6. Simplified 31.9%

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

   if -0.0 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 98.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)} \]
   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 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 25.0%

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

      1. clear-num 25.0%

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

      2. associate-/r/ 25.0%

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

      3. associate-/r* 28.6%

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

   4. Applied egg-rr 28.6%

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

      1. associate-*l/ 45.1%

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

      2. sqrt-div 50.1%

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

      3. associate-*l/ 50.0%

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

      4. *-un-lft-identity 50.0%

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

      5. clear-num 50.1%

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

      6. un-div-inv 50.0%

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

      7. sqrt-undiv 40.3%

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

      8. div-inv 40.3%

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

      9. clear-num 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-*l/ 25.0%

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

      3. associate-*r/ 40.3%

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

   8. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. sqrt-prod 50.2%

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

      3. times-frac 50.0%

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

   10. Applied egg-rr 50.0%

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

       1. associate-*r/ 50.2%

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

       2. associate-*l/ 50.1%

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

       3. *-lft-identity 50.1%

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

   12. Simplified 50.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0\_m \cdot \frac{{\left(\frac{V}{A}\right)}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 l) (- INFINITY))
    (* (/ c0_m (sqrt l)) (sqrt (/ A V)))
    (if (<= (* V l) -1e-108)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (* c0_m (/ (pow (/ V A) -0.5) (sqrt l)))
        (if (<= (* V l) 1e+255)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (/ (/ c0_m (sqrt V)) (sqrt (/ 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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m * (pow((V / A), -0.5) / sqrt(l));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0_m / sqrt(V)) / sqrt((l / A));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m * (Math.pow((V / A), -0.5) / Math.sqrt(l));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0_m / Math.sqrt(V)) / Math.sqrt((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):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -1e-108:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m * (math.pow((V / A), -0.5) / math.sqrt(l))
	elif (V * l) <= 1e+255:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0_m / math.sqrt(V)) / math.sqrt((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)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -1e-108)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m * Float64((Float64(V / A) ^ -0.5) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+255)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0_m / sqrt(V)) / sqrt(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)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -1e-108)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m * (((V / A) ^ -0.5) / sqrt(l));
	elseif ((V * l) <= 1e+255)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0_m / sqrt(V)) / sqrt((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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-108], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m * N[(N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+255], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0$95$m / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000004e-108

   1. Initial program 95.1%

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

   if -1.00000000000000004e-108 < (*.f64 V l) < -0.0

   1. Initial program 55.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div 31.9%

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

      3. div-inv 31.9%

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

   4. Applied egg-rr 31.9%

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

      1. associate-*r/ 31.9%

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

      2. *-rgt-identity 31.9%

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

   6. Simplified 31.9%

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

      1. clear-num 31.9%

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

      2. sqrt-div 32.3%

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

      3. metadata-eval 32.3%

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

   8. Applied egg-rr 32.3%

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

      1. inv-pow 32.3%

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

      2. sqrt-pow2 32.4%

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

      3. metadata-eval 32.4%

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

   10. Applied egg-rr 32.4%

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

   if -0.0 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 98.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)} \]
   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 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 25.0%

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

      1. clear-num 25.0%

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

      2. associate-/r/ 25.0%

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

      3. associate-/r* 28.6%

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

   4. Applied egg-rr 28.6%

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

      1. associate-*l/ 45.1%

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

      2. sqrt-div 50.1%

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

      3. associate-*l/ 50.0%

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

      4. *-un-lft-identity 50.0%

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

      5. clear-num 50.1%

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

      6. un-div-inv 50.0%

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

      7. sqrt-undiv 40.3%

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

      8. div-inv 40.3%

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

      9. clear-num 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-*l/ 25.0%

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

      3. associate-*r/ 40.3%

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

   8. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. sqrt-prod 50.2%

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

      3. times-frac 50.0%

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

   10. Applied egg-rr 50.0%

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

       1. associate-*r/ 50.2%

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

       2. associate-*l/ 50.1%

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

       3. *-lft-identity 50.1%

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

   12. Simplified 50.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{V}{A}\right)}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\left(c0\_m \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0\_m \cdot \frac{{\left(\frac{V}{A}\right)}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 l) -1e+202)
    (* (* c0_m (sqrt (/ A V))) (pow l -0.5))
    (if (<= (* V l) -1e-108)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (* c0_m (/ (pow (/ V A) -0.5) (sqrt l)))
        (if (<= (* V l) 1e+255)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (/ (/ c0_m (sqrt V)) (sqrt (/ 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 tmp;
	if ((V * l) <= -1e+202) {
		tmp = (c0_m * sqrt((A / V))) * pow(l, -0.5);
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m * (pow((V / A), -0.5) / sqrt(l));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0_m / sqrt(V)) / sqrt((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) :: tmp
    if ((v * l) <= (-1d+202)) then
        tmp = (c0_m * sqrt((a / v))) * (l ** (-0.5d0))
    else if ((v * l) <= (-1d-108)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0_m * (((v / a) ** (-0.5d0)) / sqrt(l))
    else if ((v * l) <= 1d+255) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = (c0_m / sqrt(v)) / sqrt((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 tmp;
	if ((V * l) <= -1e+202) {
		tmp = (c0_m * Math.sqrt((A / V))) * Math.pow(l, -0.5);
	} else if ((V * l) <= -1e-108) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m * (Math.pow((V / A), -0.5) / Math.sqrt(l));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0_m / Math.sqrt(V)) / Math.sqrt((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):
	tmp = 0
	if (V * l) <= -1e+202:
		tmp = (c0_m * math.sqrt((A / V))) * math.pow(l, -0.5)
	elif (V * l) <= -1e-108:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m * (math.pow((V / A), -0.5) / math.sqrt(l))
	elif (V * l) <= 1e+255:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0_m / math.sqrt(V)) / math.sqrt((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)
	tmp = 0.0
	if (Float64(V * l) <= -1e+202)
		tmp = Float64(Float64(c0_m * sqrt(Float64(A / V))) * (l ^ -0.5));
	elseif (Float64(V * l) <= -1e-108)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m * Float64((Float64(V / A) ^ -0.5) / sqrt(l)));
	elseif (Float64(V * l) <= 1e+255)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0_m / sqrt(V)) / sqrt(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)
	tmp = 0.0;
	if ((V * l) <= -1e+202)
		tmp = (c0_m * sqrt((A / V))) * (l ^ -0.5);
	elseif ((V * l) <= -1e-108)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m * (((V / A) ^ -0.5) / sqrt(l));
	elseif ((V * l) <= 1e+255)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0_m / sqrt(V)) / sqrt((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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+202], N[(N[(c0$95$m * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-108], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m * N[(N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+255], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0$95$m / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.999999999999999e201

   1. Initial program 48.2%

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

      1. add-sqr-sqrt 34.6%

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

      2. sqrt-unprod 26.4%

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

      3. *-commutative 26.4%

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

      4. *-commutative 26.4%

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

      5. swap-sqr 25.8%

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

      6. add-sqr-sqrt 25.8%

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

      7. pow2 25.8%

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

   4. Applied egg-rr 25.8%

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

      1. associate-/r* 25.7%

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

   6. Simplified 25.7%

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

      1. *-commutative 25.7%

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

      2. sqrt-prod 29.8%

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

      3. sqrt-pow1 61.2%

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

      4. metadata-eval 61.2%

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

      5. pow1 61.2%

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

      6. sqrt-undiv 36.7%

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

      7. associate-*r/ 36.7%

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

      8. div-inv 36.7%

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

      9. pow1/2 36.7%

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

      10. pow-flip 36.7%

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

      11. metadata-eval 36.7%

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

   8. Applied egg-rr 36.7%

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

   if -9.999999999999999e201 < (*.f64 V l) < -1.00000000000000004e-108

   1. Initial program 96.2%

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

   if -1.00000000000000004e-108 < (*.f64 V l) < -0.0

   1. Initial program 55.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div 31.9%

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

      3. div-inv 31.9%

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

   4. Applied egg-rr 31.9%

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

      1. associate-*r/ 31.9%

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

      2. *-rgt-identity 31.9%

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

   6. Simplified 31.9%

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

      1. clear-num 31.9%

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

      2. sqrt-div 32.3%

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

      3. metadata-eval 32.3%

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

   8. Applied egg-rr 32.3%

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

      1. inv-pow 32.3%

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

      2. sqrt-pow2 32.4%

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

      3. metadata-eval 32.4%

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

   10. Applied egg-rr 32.4%

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

   if -0.0 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 98.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)} \]
   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 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 25.0%

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

      1. clear-num 25.0%

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

      2. associate-/r/ 25.0%

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

      3. associate-/r* 28.6%

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

   4. Applied egg-rr 28.6%

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

      1. associate-*l/ 45.1%

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

      2. sqrt-div 50.1%

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

      3. associate-*l/ 50.0%

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

      4. *-un-lft-identity 50.0%

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

      5. clear-num 50.1%

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

      6. un-div-inv 50.0%

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

      7. sqrt-undiv 40.3%

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

      8. div-inv 40.3%

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

      9. clear-num 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-*l/ 25.0%

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

      3. associate-*r/ 40.3%

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

   8. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. sqrt-prod 50.2%

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

      3. times-frac 50.0%

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

   10. Applied egg-rr 50.0%

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

       1. associate-*r/ 50.2%

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

       2. associate-*l/ 50.1%

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

       3. *-lft-identity 50.1%

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

   12. Simplified 50.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-108}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{V}{A}\right)}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-250}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 l) (- INFINITY))
    (* (/ c0_m (sqrt l)) (sqrt (/ A V)))
    (if (<= (* V l) -4e-250)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 5e-319)
        (/ c0_m (sqrt (* V (/ l A))))
        (if (<= (* V l) 1e+255)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (/ (/ c0_m (sqrt V)) (sqrt (/ 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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -4e-250) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0_m / sqrt(V)) / sqrt((l / A));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -4e-250) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0_m / Math.sqrt(V)) / Math.sqrt((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):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -4e-250:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-319:
		tmp = c0_m / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+255:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0_m / math.sqrt(V)) / math.sqrt((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)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -4e-250)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-319)
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+255)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0_m / sqrt(V)) / sqrt(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)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -4e-250)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-319)
		tmp = c0_m / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+255)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0_m / sqrt(V)) / sqrt((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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-250], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-319], N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+255], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0$95$m / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -4.0000000000000002e-250

   1. Initial program 88.6%

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

      1. frac-2neg 88.6%

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

      2. sqrt-div 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   if -4.0000000000000002e-250 < (*.f64 V l) < 4.9999937e-319

   1. Initial program 40.0%

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

      1. clear-num 40.0%

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

      2. associate-/r/ 38.3%

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

      3. associate-/r* 38.3%

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

   4. Applied egg-rr 38.3%

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

      1. associate-*l/ 63.8%

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

      2. sqrt-div 27.2%

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

      3. associate-*l/ 27.2%

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

      4. *-un-lft-identity 27.2%

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

      5. clear-num 27.2%

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

      6. un-div-inv 27.1%

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

      7. sqrt-undiv 65.1%

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

      8. div-inv 65.1%

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

      9. clear-num 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*l/ 40.0%

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

      3. associate-*r/ 65.2%

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

   8. Simplified 65.2%

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

   if 4.9999937e-319 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 25.0%

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

      1. clear-num 25.0%

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

      2. associate-/r/ 25.0%

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

      3. associate-/r* 28.6%

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

   4. Applied egg-rr 28.6%

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

      1. associate-*l/ 45.1%

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

      2. sqrt-div 50.1%

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

      3. associate-*l/ 50.0%

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

      4. *-un-lft-identity 50.0%

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

      5. clear-num 50.1%

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

      6. un-div-inv 50.0%

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

      7. sqrt-undiv 40.3%

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

      8. div-inv 40.3%

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

      9. clear-num 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-*l/ 25.0%

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

      3. associate-*r/ 40.3%

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

   8. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. sqrt-prod 50.2%

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

      3. times-frac 50.0%

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

   10. Applied egg-rr 50.0%

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

       1. associate-*r/ 50.2%

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

       2. associate-*l/ 50.1%

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

       3. *-lft-identity 50.1%

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

   12. Simplified 50.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-250}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 l) (- INFINITY))
    (* (/ c0_m (sqrt l)) (sqrt (/ A V)))
    (if (<= (* V l) -1e-295)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 5e-319)
        (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
        (if (<= (* V l) 1e+255)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (/ (/ c0_m (sqrt V)) (sqrt (/ 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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -1e-295) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0_m / sqrt(V)) / sqrt((l / A));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0_m / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -1e-295) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= 1e+255) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0_m / Math.sqrt(V)) / Math.sqrt((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):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (c0_m / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -1e-295:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-319:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= 1e+255:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0_m / math.sqrt(V)) / math.sqrt((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)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(c0_m / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -1e-295)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-319)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 1e+255)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0_m / sqrt(V)) / sqrt(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)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (c0_m / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -1e-295)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-319)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= 1e+255)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0_m / sqrt(V)) / sqrt((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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-295], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-319], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+255], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0$95$m / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 25.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. sqrt-unprod 25.7%

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

      3. *-commutative 25.7%

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

      4. *-commutative 25.7%

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

      5. swap-sqr 24.8%

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

      6. add-sqr-sqrt 24.8%

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

      7. pow2 24.8%

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

   4. Applied egg-rr 24.8%

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

      1. associate-/r* 24.7%

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

   6. Simplified 24.7%

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

      1. sqrt-prod 31.4%

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

      2. sqrt-undiv 7.3%

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

      3. sqrt-undiv 31.4%

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

      4. associate-/l/ 24.8%

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

      5. frac-2neg 24.8%

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

      6. distribute-rgt-neg-out 24.8%

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

      7. sqrt-undiv 24.8%

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

      8. sqrt-pow1 25.7%

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

      9. metadata-eval 25.7%

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

      10. pow1 25.7%

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

      11. associate-*l/ 25.7%

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

      12. *-commutative 25.7%

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

      13. sqrt-prod 21.2%

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

      14. times-frac 21.3%

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

      15. sqrt-div 14.8%

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

      16. frac-2neg 14.8%

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

   8. Applied egg-rr 14.8%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000006e-295

   1. Initial program 89.0%

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

      1. frac-2neg 89.0%

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

      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 -1.00000000000000006e-295 < (*.f64 V l) < 4.9999937e-319

   1. Initial program 32.7%

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

      1. clear-num 32.7%

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

      2. associate-/r/ 30.9%

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

      3. associate-/r* 30.9%

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

   4. Applied egg-rr 30.9%

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

      1. associate-*l/ 59.5%

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

      2. associate-*r/ 59.4%

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

      3. associate-*l/ 59.5%

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

      4. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. frac-2neg 59.5%

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

      2. sqrt-div 42.5%

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

      3. distribute-neg-frac2 42.5%

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

   8. Applied egg-rr 42.5%

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

   if 4.9999937e-319 < (*.f64 V l) < 9.99999999999999988e254

   1. Initial program 87.4%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if 9.99999999999999988e254 < (*.f64 V l)

   1. Initial program 25.0%

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

      1. clear-num 25.0%

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

      2. associate-/r/ 25.0%

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

      3. associate-/r* 28.6%

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

   4. Applied egg-rr 28.6%

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

      1. associate-*l/ 45.1%

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

      2. sqrt-div 50.1%

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

      3. associate-*l/ 50.0%

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

      4. *-un-lft-identity 50.0%

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

      5. clear-num 50.1%

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

      6. un-div-inv 50.0%

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

      7. sqrt-undiv 40.3%

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

      8. div-inv 40.3%

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

      9. clear-num 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-*l/ 25.0%

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

      3. associate-*r/ 40.3%

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

   8. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. sqrt-prod 50.2%

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

      3. times-frac 50.0%

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

   10. Applied egg-rr 50.0%

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

       1. associate-*r/ 50.2%

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

       2. associate-*l/ 50.1%

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

       3. *-lft-identity 50.1%

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

   12. Simplified 50.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-295}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{V}}}{\sqrt{\frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-179}:\\ \;\;\;\;c0\_m \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #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) -2e-179)
    (* c0_m (sqrt (* A (/ (/ 1.0 V) l))))
    (if (<= (* V l) 5e-319)
      (/ c0_m (sqrt (* V (/ l A))))
      (* c0_m (/ (sqrt A) (sqrt (* V l))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-179) {
		tmp = c0_m * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m / sqrt((V * (l / A)));
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-179)) then
        tmp = c0_m * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 5d-319) then
        tmp = c0_m / sqrt((v * (l / a)))
    else
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-179) {
		tmp = c0_m * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 5e-319) {
		tmp = c0_m / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -2e-179:
		tmp = c0_m * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 5e-319:
		tmp = c0_m / math.sqrt((V * (l / A)))
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-179)
		tmp = Float64(c0_m * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 5e-319)
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-179)
		tmp = c0_m * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 5e-319)
		tmp = c0_m / sqrt((V * (l / A)));
	else
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -2e-179], N[(c0$95$m * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-319], N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -2e-179

   1. Initial program 83.5%

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

      1. clear-num 83.4%

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

      2. associate-/r/ 83.4%

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

      3. associate-/r* 85.0%

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

   4. Applied egg-rr 85.0%

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

   if -2e-179 < (*.f64 V l) < 4.9999937e-319

   1. Initial program 45.4%

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

      1. clear-num 45.3%

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

      2. associate-/r/ 44.1%

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

      3. associate-/r* 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. associate-*l/ 62.6%

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

      2. sqrt-div 27.6%

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

      3. associate-*l/ 27.6%

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

      4. *-un-lft-identity 27.6%

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

      5. clear-num 27.6%

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

      6. un-div-inv 27.6%

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

      7. sqrt-undiv 63.6%

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

      8. div-inv 63.6%

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

      9. clear-num 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l/ 45.4%

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

      3. associate-*r/ 61.8%

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

   8. Simplified 61.8%

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

   if 4.9999937e-319 < (*.f64 V l)

   1. Initial program 78.5%

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

      1. sqrt-div 88.8%

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

      2. div-inv 88.7%

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

   4. Applied egg-rr 88.7%

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

      1. associate-*r/ 88.8%

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

      2. *-rgt-identity 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-179}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+306)))
      (* c0_m (sqrt (/ (/ A V) l)))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+306))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+306):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+306)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+306]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 29.8%

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

      1. associate-/r* 44.5%

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

   3. Simplified 44.5%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.5% 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 73.7%

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

3. Final simplification 73.7%

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

Reproduce

herbie shell --seed 2024072 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))