Henrywood and Agarwal, Equation (3)

Percentage Accurate: 75.1% → 91.6%

Time: 10.9s

Alternatives: 11

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: 75.1% 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: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{A}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-305}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t\_0}{\ell} \cdot \frac{t\_0}{V}}\\ \end{array} \end{array} \end{array} \]

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 (/ V A)))
      (if (<= (* V l) -2e-305)
        (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
        (if (<= (* V l) 2e-304)
          (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
          (if (<= (* V l) 1e+307)
            (* 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((V / A));
	} else if ((V * l) <= -2e-305) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-304) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= 1e+307) {
		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((V / A));
	} else if ((V * l) <= -2e-305) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-304) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= 1e+307) {
		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((V / A))
	elif (V * l) <= -2e-305:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-304:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= 1e+307:
		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(V / A)));
	elseif (Float64(V * l) <= -2e-305)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-304)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 1e+307)
		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((V / A));
	elseif ((V * l) <= -2e-305)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-304)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= 1e+307)
		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[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-305], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-304], 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+307], 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 47.3%

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

      1. add-sqr-sqrt 47.3%

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

      2. sqrt-unprod 47.3%

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

      3. *-commutative 47.3%

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

      4. *-commutative 47.3%

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

      5. swap-sqr 47.3%

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

      6. add-sqr-sqrt 47.3%

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

      7. pow2 47.3%

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

   4. Applied egg-rr 47.3%

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

      1. associate-/r* 56.8%

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

   6. Simplified 56.8%

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

      1. sqrt-prod 56.8%

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

      2. sqrt-undiv 29.8%

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

      3. sqrt-pow1 29.8%

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

      4. metadata-eval 29.8%

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

      5. pow1 29.8%

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

      6. *-commutative 29.8%

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

      7. add-cbrt-cube 29.8%

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

      8. unpow2 29.8%

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

      9. cbrt-prod 29.8%

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

      10. associate-*l* 29.8%

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

      11. unpow2 29.8%

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

      12. cbrt-prod 29.8%

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

      13. pow2 29.8%

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

      14. sqrt-undiv 66.5%

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

      15. associate-/r* 47.3%

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

   8. Applied egg-rr 47.3%

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

      1. associate-*r* 47.3%

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

      2. unpow2 47.3%

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

      3. add-cube-cbrt 47.3%

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

      4. sqrt-div 0.0%

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

      5. clear-num 0.0%

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

      6. sqrt-div 47.3%

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

      7. *-commutative 47.3%

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

      8. associate-*r/ 66.3%

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

      9. un-div-inv 66.5%

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

      10. sqrt-prod 30.0%

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

      11. associate-/r* 29.8%

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

   10. Applied egg-rr 29.8%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-305

   1. Initial program 80.1%

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

      1. frac-2neg 80.1%

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

      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 -1.99999999999999999e-305 < (*.f64 V l) < 1.99999999999999994e-304

   1. Initial program 45.0%

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

      1. clear-num 45.0%

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

      2. associate-/r/ 45.0%

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

      3. associate-/r* 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. associate-/l/ 45.0%

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

      2. associate-*l/ 45.0%

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

      3. *-un-lft-identity 45.0%

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

      4. associate-/r* 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. frac-2neg 56.6%

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

      2. sqrt-div 45.4%

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

      3. distribute-neg-frac2 45.4%

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

   8. Applied egg-rr 45.4%

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

   if 1.99999999999999994e-304 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 88.0%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 23.4%

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

      1. add-sqr-sqrt 23.4%

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

      2. sqrt-unprod 23.4%

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

      3. *-commutative 23.4%

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

      4. *-commutative 23.4%

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

      5. swap-sqr 22.5%

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

      6. add-sqr-sqrt 22.5%

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

      7. pow2 22.5%

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

   4. Applied egg-rr 22.5%

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

      1. associate-/r* 27.1%

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

   6. Simplified 27.1%

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

      1. associate-*l/ 36.8%

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

      2. *-commutative 36.8%

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

      3. associate-*r/ 27.1%

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

      4. associate-/r* 22.5%

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

      5. associate-*r/ 22.4%

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

   8. Applied egg-rr 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. *-commutative 22.4%

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

      3. times-frac 37.0%

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

      4. sqrt-prod 37.0%

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

      5. sqrt-pow1 36.6%

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

      6. metadata-eval 36.6%

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

      7. pow1 36.6%

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

      8. sqrt-prod 36.6%

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

      9. sqrt-pow1 51.7%

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

      10. metadata-eval 51.7%

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

      11. pow1 51.7%

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

   10. Applied egg-rr 51.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \sqrt{A}}{\ell} \cdot \frac{c0 \cdot \sqrt{A}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-209}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 4e-209)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 2e+287) t_0 (/ 1.0 (/ (sqrt (* V (/ l A))) c0_m)))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-209) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+287) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0_m);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 4d-209) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 2d+287) then
        tmp = t_0
    else
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0_m)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-209) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+287) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0_m);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-209:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+287:
		tmp = t_0
	else:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0_m)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-209)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+287)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0_m));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-209)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+287)
		tmp = t_0;
	else
		tmp = 1.0 / (sqrt((V * (l / A))) / c0_m);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 4e-209], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+287], t$95$0, N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e-209

   1. Initial program 69.9%

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

      1. clear-num 69.9%

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

      2. associate-/r/ 69.9%

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

      3. associate-/r* 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. associate-/l/ 69.9%

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

      2. associate-*l/ 69.9%

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

      3. *-un-lft-identity 69.9%

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

      4. associate-/r* 67.0%

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

   6. Applied egg-rr 67.0%

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

   if 4.0000000000000002e-209 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000002e287

   1. Initial program 98.1%

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

   if 2.0000000000000002e287 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 43.4%

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

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 35.1%

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

      3. *-commutative 35.1%

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

      4. *-commutative 35.1%

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

      5. swap-sqr 34.5%

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

      6. add-sqr-sqrt 34.5%

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

      7. pow2 34.5%

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

   4. Applied egg-rr 34.5%

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

      1. associate-/r* 43.6%

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

   6. Simplified 43.6%

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

      1. sqrt-prod 43.5%

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

      2. sqrt-undiv 26.0%

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

      3. sqrt-pow1 32.1%

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

      4. metadata-eval 32.1%

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

      5. pow1 32.1%

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

      6. *-commutative 32.1%

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

      7. add-cbrt-cube 23.1%

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

      8. unpow2 23.1%

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

      9. cbrt-prod 26.0%

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

      10. associate-*l* 26.1%

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

      11. unpow2 26.1%

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

      12. cbrt-prod 31.7%

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

      13. pow2 31.7%

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

      14. sqrt-undiv 52.1%

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

      15. associate-/r* 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. associate-*r* 43.4%

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

      2. unpow2 43.4%

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

      3. add-cube-cbrt 43.4%

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

      4. sqrt-div 25.5%

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

      5. clear-num 25.4%

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

      6. sqrt-div 44.2%

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

      7. *-commutative 44.2%

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

      8. associate-*r/ 53.1%

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

      9. un-div-inv 53.2%

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

      10. clear-num 53.3%

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

      11. *-commutative 53.3%

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

      12. associate-/r/ 50.4%

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

      13. div-inv 50.5%

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

      14. clear-num 50.5%

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

   10. Applied egg-rr 50.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-209}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-209}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 4e-209)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 2e+287) 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 <= 4e-209) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+287) {
		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 <= 4d-209) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 2d+287) 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 <= 4e-209) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+287) {
		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 <= 4e-209:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+287:
		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 <= 4e-209)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+287)
		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 <= 4e-209)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+287)
		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, 4e-209], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+287], 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)))) < 4.0000000000000002e-209

   1. Initial program 69.9%

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

      1. clear-num 69.9%

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

      2. associate-/r/ 69.9%

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

      3. associate-/r* 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. associate-/l/ 69.9%

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

      2. associate-*l/ 69.9%

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

      3. *-un-lft-identity 69.9%

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

      4. associate-/r* 67.0%

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

   6. Applied egg-rr 67.0%

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

   if 4.0000000000000002e-209 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000002e287

   1. Initial program 98.1%

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

   if 2.0000000000000002e287 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 43.4%

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

      1. clear-num 43.4%

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

      2. associate-/r/ 43.4%

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

      3. associate-/r* 43.4%

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

   4. Applied egg-rr 43.4%

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

      1. associate-*l/ 52.3%

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

      2. sqrt-div 32.1%

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

      3. associate-*l/ 32.1%

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

      4. *-un-lft-identity 32.1%

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

      5. clear-num 32.1%

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

      6. un-div-inv 32.2%

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

      7. sqrt-undiv 53.3%

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

      8. div-inv 53.2%

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

      9. clear-num 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. *-commutative 53.2%

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

      2. associate-*l/ 44.3%

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

      3. associate-/l* 50.5%

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

   8. Simplified 50.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-209}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-145}:\\ \;\;\;\;c0\_m \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{c0\_m}^{2} \cdot \frac{A}{\ell}}{V}}\\ \end{array} \end{array} \end{array} \]

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 V)) (sqrt l)))))
   (*
    c0_s
    (if (<= (* V l) -2e+152)
      t_0
      (if (<= (* V l) -5e-145)
        (* c0_m (sqrt (* A (/ (/ 1.0 V) l))))
        (if (<= (* V l) 0.0)
          t_0
          (if (<= (* V l) 1e+307)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (sqrt (/ (* (pow c0_m 2.0) (/ A l)) 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 / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+152) {
		tmp = t_0;
	} else if ((V * l) <= -5e-145) {
		tmp = c0_m * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((pow(c0_m, 2.0) * (A / l)) / V));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-2d+152)) then
        tmp = t_0
    else if ((v * l) <= (-5d-145)) then
        tmp = c0_m * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt((((c0_m ** 2.0d0) * (a / l)) / 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 = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -2e+152) {
		tmp = t_0;
	} else if ((V * l) <= -5e-145) {
		tmp = c0_m * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((Math.pow(c0_m, 2.0) * (A / l)) / 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 / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -2e+152:
		tmp = t_0
	elif (V * l) <= -5e-145:
		tmp = c0_m * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((math.pow(c0_m, 2.0) * (A / l)) / 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 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -2e+152)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-145)
		tmp = Float64(c0_m * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64((c0_m ^ 2.0) * Float64(A / l)) / 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 / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -2e+152)
		tmp = t_0;
	elseif ((V * l) <= -5e-145)
		tmp = c0_m * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((((c0_m ^ 2.0) * (A / l)) / 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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -2e+152], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-145], 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], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Power[c0$95$m, 2.0], $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.0000000000000001e152 or -4.9999999999999998e-145 < (*.f64 V l) < 0.0

   1. Initial program 60.1%

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

      1. associate-/r* 66.3%

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

      2. sqrt-div 38.2%

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

      3. div-inv 38.1%

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

   4. Applied egg-rr 38.1%

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

      1. associate-*r/ 38.2%

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

      2. *-rgt-identity 38.2%

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

   6. Simplified 38.2%

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

   if -2.0000000000000001e152 < (*.f64 V l) < -4.9999999999999998e-145

   1. Initial program 86.9%

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

      1. clear-num 85.9%

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

      2. associate-/r/ 86.9%

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

      3. associate-/r* 86.9%

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

   4. Applied egg-rr 86.9%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 88.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)} \]
   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 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 23.4%

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

      1. add-sqr-sqrt 23.4%

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

      2. sqrt-unprod 23.4%

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

      3. *-commutative 23.4%

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

      4. *-commutative 23.4%

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

      5. swap-sqr 22.5%

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

      6. add-sqr-sqrt 22.5%

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

      7. pow2 22.5%

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

   4. Applied egg-rr 22.5%

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

      1. associate-/r* 27.1%

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

   6. Simplified 27.1%

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

      1. div-inv 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l/ 27.1%

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

      4. associate-*r/ 36.7%

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

      5. un-div-inv 36.7%

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

   8. Applied egg-rr 36.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+152}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-145}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{c0}^{2} \cdot \frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-316}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0\_m \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{c0\_m}^{2} \cdot \frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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 (/ V A)))
    (if (<= (* V l) -1e-316)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 0.0)
        (* (sqrt (/ A V)) (* c0_m (pow l -0.5)))
        (if (<= (* V l) 1e+307)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (/ (* (pow c0_m 2.0) (/ A l)) 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((V / A));
	} else if ((V * l) <= -1e-316) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt((A / V)) * (c0_m * pow(l, -0.5));
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((pow(c0_m, 2.0) * (A / l)) / 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((V / A));
	} else if ((V * l) <= -1e-316) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt((A / V)) * (c0_m * Math.pow(l, -0.5));
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((Math.pow(c0_m, 2.0) * (A / l)) / 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((V / A))
	elif (V * l) <= -1e-316:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt((A / V)) * (c0_m * math.pow(l, -0.5))
	elif (V * l) <= 1e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((math.pow(c0_m, 2.0) * (A / l)) / 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(V / A)));
	elseif (Float64(V * l) <= -1e-316)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0_m * (l ^ -0.5)));
	elseif (Float64(V * l) <= 1e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64((c0_m ^ 2.0) * Float64(A / l)) / 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((V / A));
	elseif ((V * l) <= -1e-316)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt((A / V)) * (c0_m * (l ^ -0.5));
	elseif ((V * l) <= 1e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((((c0_m ^ 2.0) * (A / l)) / 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[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-316], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0$95$m * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Power[c0$95$m, 2.0], $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 47.3%

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

      1. add-sqr-sqrt 47.3%

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

      2. sqrt-unprod 47.3%

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

      3. *-commutative 47.3%

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

      4. *-commutative 47.3%

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

      5. swap-sqr 47.3%

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

      6. add-sqr-sqrt 47.3%

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

      7. pow2 47.3%

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

   4. Applied egg-rr 47.3%

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

      1. associate-/r* 56.8%

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

   6. Simplified 56.8%

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

      1. sqrt-prod 56.8%

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

      2. sqrt-undiv 29.8%

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

      3. sqrt-pow1 29.8%

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

      4. metadata-eval 29.8%

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

      5. pow1 29.8%

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

      6. *-commutative 29.8%

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

      7. add-cbrt-cube 29.8%

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

      8. unpow2 29.8%

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

      9. cbrt-prod 29.8%

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

      10. associate-*l* 29.8%

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

      11. unpow2 29.8%

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

      12. cbrt-prod 29.8%

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

      13. pow2 29.8%

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

      14. sqrt-undiv 66.5%

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

      15. associate-/r* 47.3%

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

   8. Applied egg-rr 47.3%

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

      1. associate-*r* 47.3%

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

      2. unpow2 47.3%

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

      3. add-cube-cbrt 47.3%

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

      4. sqrt-div 0.0%

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

      5. clear-num 0.0%

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

      6. sqrt-div 47.3%

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

      7. *-commutative 47.3%

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

      8. associate-*r/ 66.3%

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

      9. un-div-inv 66.5%

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

      10. sqrt-prod 30.0%

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

      11. associate-/r* 29.8%

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

   10. Applied egg-rr 29.8%

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

   if -inf.0 < (*.f64 V l) < -9.999999837e-317

   1. Initial program 79.5%

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

      1. frac-2neg 79.5%

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

      2. sqrt-div 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. distribute-rgt-neg-out 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

   if -9.999999837e-317 < (*.f64 V l) < 0.0

   1. Initial program 42.8%

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

      1. add-sqr-sqrt 17.8%

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

      2. sqrt-unprod 18.0%

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

      3. *-commutative 18.0%

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

      4. *-commutative 18.0%

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

      5. swap-sqr 17.7%

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

      6. add-sqr-sqrt 17.7%

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

      7. pow2 17.7%

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

   4. Applied egg-rr 17.7%

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

      1. associate-/r* 27.5%

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

   6. Simplified 27.5%

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

      1. sqrt-prod 27.3%

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

      2. div-inv 27.3%

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

      3. sqrt-prod 16.9%

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

      4. sqrt-pow1 36.7%

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

      5. metadata-eval 36.7%

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

      6. pow1 36.7%

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

      7. associate-*l* 36.9%

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

      8. inv-pow 36.9%

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

      9. sqrt-pow1 36.9%

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

      10. metadata-eval 36.9%

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

   8. Applied egg-rr 36.9%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 88.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)} \]
   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 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 23.4%

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

      1. add-sqr-sqrt 23.4%

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

      2. sqrt-unprod 23.4%

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

      3. *-commutative 23.4%

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

      4. *-commutative 23.4%

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

      5. swap-sqr 22.5%

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

      6. add-sqr-sqrt 22.5%

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

      7. pow2 22.5%

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

   4. Applied egg-rr 22.5%

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

      1. associate-/r* 27.1%

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

   6. Simplified 27.1%

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

      1. div-inv 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l/ 27.1%

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

      4. associate-*r/ 36.7%

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

      5. un-div-inv 36.7%

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

   8. Applied egg-rr 36.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{c0}^{2} \cdot \frac{A}{\ell}}{V}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{c0}^{2} \cdot \frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0\_m}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-305}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{c0\_m}^{2} \cdot \frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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 (/ V A)))
    (if (<= (* V l) -2e-305)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 2e-304)
        (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
        (if (<= (* V l) 1e+307)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (/ (* (pow c0_m 2.0) (/ A l)) 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((V / A));
	} else if ((V * l) <= -2e-305) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-304) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((pow(c0_m, 2.0) * (A / l)) / 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((V / A));
	} else if ((V * l) <= -2e-305) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-304) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((Math.pow(c0_m, 2.0) * (A / l)) / 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((V / A))
	elif (V * l) <= -2e-305:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-304:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= 1e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((math.pow(c0_m, 2.0) * (A / l)) / 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(V / A)));
	elseif (Float64(V * l) <= -2e-305)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-304)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 1e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64((c0_m ^ 2.0) * Float64(A / l)) / 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((V / A));
	elseif ((V * l) <= -2e-305)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-304)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= 1e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((((c0_m ^ 2.0) * (A / l)) / 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[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-305], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-304], 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+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Power[c0$95$m, 2.0], $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 47.3%

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

      1. add-sqr-sqrt 47.3%

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

      2. sqrt-unprod 47.3%

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

      3. *-commutative 47.3%

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

      4. *-commutative 47.3%

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

      5. swap-sqr 47.3%

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

      6. add-sqr-sqrt 47.3%

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

      7. pow2 47.3%

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

   4. Applied egg-rr 47.3%

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

      1. associate-/r* 56.8%

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

   6. Simplified 56.8%

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

      1. sqrt-prod 56.8%

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

      2. sqrt-undiv 29.8%

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

      3. sqrt-pow1 29.8%

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

      4. metadata-eval 29.8%

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

      5. pow1 29.8%

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

      6. *-commutative 29.8%

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

      7. add-cbrt-cube 29.8%

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

      8. unpow2 29.8%

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

      9. cbrt-prod 29.8%

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

      10. associate-*l* 29.8%

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

      11. unpow2 29.8%

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

      12. cbrt-prod 29.8%

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

      13. pow2 29.8%

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

      14. sqrt-undiv 66.5%

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

      15. associate-/r* 47.3%

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

   8. Applied egg-rr 47.3%

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

      1. associate-*r* 47.3%

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

      2. unpow2 47.3%

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

      3. add-cube-cbrt 47.3%

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

      4. sqrt-div 0.0%

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

      5. clear-num 0.0%

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

      6. sqrt-div 47.3%

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

      7. *-commutative 47.3%

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

      8. associate-*r/ 66.3%

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

      9. un-div-inv 66.5%

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

      10. sqrt-prod 30.0%

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

      11. associate-/r* 29.8%

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

   10. Applied egg-rr 29.8%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-305

   1. Initial program 80.1%

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

      1. frac-2neg 80.1%

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

      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 -1.99999999999999999e-305 < (*.f64 V l) < 1.99999999999999994e-304

   1. Initial program 45.0%

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

      1. clear-num 45.0%

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

      2. associate-/r/ 45.0%

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

      3. associate-/r* 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. associate-/l/ 45.0%

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

      2. associate-*l/ 45.0%

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

      3. *-un-lft-identity 45.0%

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

      4. associate-/r* 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. frac-2neg 56.6%

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

      2. sqrt-div 45.4%

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

      3. distribute-neg-frac2 45.4%

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

   8. Applied egg-rr 45.4%

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

   if 1.99999999999999994e-304 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 88.0%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 23.4%

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

      1. add-sqr-sqrt 23.4%

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

      2. sqrt-unprod 23.4%

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

      3. *-commutative 23.4%

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

      4. *-commutative 23.4%

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

      5. swap-sqr 22.5%

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

      6. add-sqr-sqrt 22.5%

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

      7. pow2 22.5%

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

   4. Applied egg-rr 22.5%

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

      1. associate-/r* 27.1%

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

   6. Simplified 27.1%

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

      1. div-inv 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l/ 27.1%

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

      4. associate-*r/ 36.7%

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

      5. un-div-inv 36.7%

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

   8. Applied egg-rr 36.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{c0}^{2} \cdot \frac{A}{\ell}}{V}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{c0}^{2} \cdot \frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-145}:\\ \;\;\;\;c0\_m \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \end{array} \]

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) -2e+152)
      t_0
      (if (<= (* V l) -5e-145)
        (* c0_m (sqrt (* A (/ (/ 1.0 V) l))))
        (if (<= (* V l) 0.0)
          t_0
          (if (<= (* V l) 1e+307)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (* 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) {
	double t_0 = c0_m * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+152) {
		tmp = t_0;
	} else if ((V * l) <= -5e-145) {
		tmp = c0_m * sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m * sqrt(((A / 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) <= (-2d+152)) then
        tmp = t_0
    else if ((v * l) <= (-5d-145)) then
        tmp = c0_m * sqrt((a * ((1.0d0 / v) / l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0_m * sqrt(((a / 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) <= -2e+152) {
		tmp = t_0;
	} else if ((V * l) <= -5e-145) {
		tmp = c0_m * Math.sqrt((A * ((1.0 / V) / l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m * Math.sqrt(((A / 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) <= -2e+152:
		tmp = t_0
	elif (V * l) <= -5e-145:
		tmp = c0_m * math.sqrt((A * ((1.0 / V) / l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m * math.sqrt(((A / 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) <= -2e+152)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-145)
		tmp = Float64(c0_m * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / 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) <= -2e+152)
		tmp = t_0;
	elseif ((V * l) <= -5e-145)
		tmp = c0_m * sqrt((A * ((1.0 / V) / l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m * sqrt(((A / 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], -2e+152], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-145], 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], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.0000000000000001e152 or -4.9999999999999998e-145 < (*.f64 V l) < 0.0

   1. Initial program 60.1%

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

      1. associate-/r* 66.3%

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

      2. sqrt-div 38.2%

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

      3. div-inv 38.1%

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

   4. Applied egg-rr 38.1%

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

      1. associate-*r/ 38.2%

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

      2. *-rgt-identity 38.2%

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

   6. Simplified 38.2%

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

   if -2.0000000000000001e152 < (*.f64 V l) < -4.9999999999999998e-145

   1. Initial program 86.9%

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

      1. clear-num 85.9%

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

      2. associate-/r/ 86.9%

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

      3. associate-/r* 86.9%

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

   4. Applied egg-rr 86.9%

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

   if 0.0 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 88.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)} \]
   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 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 23.4%

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

      1. associate-/r* 49.1%

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

   3. Simplified 49.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+152}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-145}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-145}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -5e-145)
    (* c0_m (sqrt (/ A (* V l))))
    (if (<= (* V l) 2e-304)
      (/ c0_m (sqrt (* l (/ V A))))
      (if (<= (* V l) 1e+307)
        (* c0_m (/ (sqrt A) (sqrt (* V l))))
        (* 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) {
	double tmp;
	if ((V * l) <= -5e-145) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 2e-304) {
		tmp = c0_m / sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m * sqrt(((A / V) / l));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d-145)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 2d-304) then
        tmp = c0_m / sqrt((l * (v / a)))
    else if ((v * l) <= 1d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0_m * sqrt(((a / v) / l))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-145) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 2e-304) {
		tmp = c0_m / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -5e-145:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 2e-304:
		tmp = c0_m / math.sqrt((l * (V / A)))
	elif (V * l) <= 1e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m * math.sqrt(((A / V) / l))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e-145)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 2e-304)
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 1e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e-145)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 2e-304)
		tmp = c0_m / sqrt((l * (V / A)));
	elseif ((V * l) <= 1e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m * sqrt(((A / V) / l));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e-145], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-304], N[(c0$95$m / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e-145

   1. Initial program 77.6%

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

   if -4.9999999999999998e-145 < (*.f64 V l) < 1.99999999999999994e-304

   1. Initial program 57.4%

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

      1. clear-num 57.4%

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

      2. associate-/r/ 57.3%

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

      3. associate-/r* 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. associate-*l/ 64.2%

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

      2. sqrt-div 36.0%

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

      3. associate-*l/ 36.0%

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

      4. *-un-lft-identity 36.0%

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

      5. clear-num 36.0%

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

      6. un-div-inv 36.0%

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

      7. sqrt-undiv 64.7%

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

      8. div-inv 64.7%

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

      9. clear-num 64.7%

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

   6. Applied egg-rr 64.7%

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

   if 1.99999999999999994e-304 < (*.f64 V l) < 9.99999999999999986e306

   1. Initial program 88.0%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 9.99999999999999986e306 < (*.f64 V l)

   1. Initial program 23.4%

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

      1. associate-/r* 49.1%

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

   3. Simplified 49.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-145}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.1% accurate, 0.5× speedup

Math

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

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (* c0_s (if (<= t_0 0.0) (* c0_m (sqrt (/ (/ A V) l))) t_0))))

C

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

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

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

Python

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

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 67.7%

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

      1. associate-/r* 67.9%

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

   3. Simplified 67.9%

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

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

   1. Initial program 80.7%

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

4. Final simplification 72.9%

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

Alternative 10: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-209}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (* c0_s (if (<= t_0 4e-209) (* c0_m (sqrt (/ (/ A l) V))) t_0))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-209) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 4d-209) then
        tmp = c0_m * sqrt(((a / l) / v))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-209) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-209:
		tmp = c0_m * math.sqrt(((A / l) / V))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-209)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-209)
		tmp = c0_m * sqrt(((A / l) / V));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 4e-209], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e-209

   1. Initial program 69.9%

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

      1. clear-num 69.9%

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

      2. associate-/r/ 69.9%

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

      3. associate-/r* 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. associate-/l/ 69.9%

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

      2. associate-*l/ 69.9%

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

      3. *-un-lft-identity 69.9%

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

      4. associate-/r* 67.0%

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

   6. Applied egg-rr 67.0%

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

   if 4.0000000000000002e-209 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 78.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-209}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% 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 72.8%

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

3. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2024081 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))