Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 91.9%

Time: 10.7s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{0 - V}\\ t_1 := \sqrt{0 - A}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{t\_0} \cdot t\_1\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-255}:\\ \;\;\;\;c0 \cdot \frac{t\_1}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - \frac{A}{\ell}}}{t\_0}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.0 V))) (t_1 (sqrt (- 0.0 A))))
   (if (<= (* V l) (- INFINITY))
     (* (/ (/ c0 (sqrt l)) t_0) t_1)
     (if (<= (* V l) -5e-255)
       (* c0 (/ t_1 (sqrt (- 0.0 (fma l V 0.0)))))
       (if (<= (* V l) 0.0)
         (* c0 (/ (sqrt (- 0.0 (/ A l))) t_0))
         (if (<= (* V l) 1e+289)
           (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
           (* c0 (sqrt (/ (/ A V) l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((0.0 - V));
	double t_1 = sqrt((0.0 - A));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = ((c0 / sqrt(l)) / t_0) * t_1;
	} else if ((V * l) <= -5e-255) {
		tmp = c0 * (t_1 / sqrt((0.0 - fma(l, V, 0.0))));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((0.0 - (A / l))) / t_0);
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(0.0 - V))
	t_1 = sqrt(Float64(0.0 - A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c0 / sqrt(l)) / t_0) * t_1);
	elseif (Float64(V * l) <= -5e-255)
		tmp = Float64(c0 * Float64(t_1 / sqrt(Float64(0.0 - fma(l, V, 0.0)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - Float64(A / l))) / t_0));
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-255], N[(c0 * N[(t$95$1 / N[Sqrt[N[(0.0 - N[(l * V + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(0.0 - N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 38.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 52.7

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

   4. Applied egg-rr 52.7%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. associate-*r/ N/A

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

      8. pow1/2 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. un-div-inv N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

      14. pow1/2 N/A

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

      15. associate-/r/ N/A

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

      16. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{0 - V}} \cdot \sqrt{0 - A}} \]

   if -inf.0 < (*.f64 V l) < -4.9999999999999996e-255

   1. Initial program 86.7%

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

      1. remove-double-div N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. inv-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. sqr-neg N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-pow N/A

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

      18. inv-pow N/A

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

      19. inv-pow N/A

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

      20. frac-2neg N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. clear-num N/A

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

      24. associate-/r* N/A

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

      25. neg-mul-1 N/A

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

      26. distribute-lft-neg-in N/A

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

   4. Applied egg-rr 79.9%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}} \]

      13. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}} \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      15. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      16. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      17. +-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\left(V \cdot \ell + 0\right)}}} \]

      18. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \left(\color{blue}{\ell \cdot V} + 0\right)}} \]

      19. accelerator-lowering-fma.f64 99.3

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\mathsf{fma}\left(\ell, V, 0\right)}}} \]

   6. Applied egg-rr 99.3%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}} \]

   if -4.9999999999999996e-255 < (*.f64 V l) < 0.0

   1. Initial program 35.9%

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

      1. remove-double-div N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. inv-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. sqr-neg N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-pow N/A

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

      18. inv-pow N/A

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

      19. inv-pow N/A

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

      20. frac-2neg N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. clear-num N/A

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

      24. associate-/r* N/A

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

      25. neg-mul-1 N/A

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

      26. distribute-lft-neg-in N/A

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

   4. Applied egg-rr 57.0%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. associate-*l/ N/A

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

      6. sqrt-div N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{\color{blue}{0 - \ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      14. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{\color{blue}{0 - \ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      16. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{0 - V}}} \]

      17. --lowering--.f64 41.1

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{0 - V}}} \]

   6. Applied egg-rr 41.1%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{0 - V}}} \]
   7. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(V\right)}}} \]

      2. neg-lowering-neg.f64 41.1

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{-V}}} \]

   8. Applied egg-rr 41.1%

      \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{-V}}} \]

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{0 - V}} \cdot \sqrt{0 - A}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - \frac{A}{\ell}}}{\sqrt{0 - V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+230}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 1e+230) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+230) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+230) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+230) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+230:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+230)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+230)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+230], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1.0000000000000001e230 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 66.6

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

   4. Applied egg-rr 66.6%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.0000000000000001e230

   1. Initial program 99.2%

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

Alternative 3: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+262}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - \frac{A}{\ell}}}{\sqrt{0 - V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+262)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -5e-255)
     (* c0 (/ (sqrt (- 0.0 A)) (sqrt (- 0.0 (fma l V 0.0)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (- 0.0 (/ A l))) (sqrt (- 0.0 V))))
       (if (<= (* V l) 1e+289)
         (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+262) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -5e-255) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - fma(l, V, 0.0))));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((0.0 - (A / l))) / sqrt((0.0 - V)));
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+262)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -5e-255)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(0.0 - fma(l, V, 0.0)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - Float64(A / l))) / sqrt(Float64(0.0 - V))));
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+262], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-255], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - N[(l * V + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(0.0 - N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1e262

   1. Initial program 50.7%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 62.1

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

   4. Applied egg-rr 62.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\mathsf{neg}\left(V\right)}{\frac{A}{\color{blue}{0 - \ell}}}}} \]

      6. sqrt-undiv N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\mathsf{neg}\left(V\right)}}{\sqrt{\frac{A}{0 - \ell}}}}} \]

      7. sub0-neg N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{\color{blue}{0 - V}}}{\sqrt{\frac{A}{0 - \ell}}}} \]

      8. sqrt-undiv N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{0 - V}}{\color{blue}{\frac{\sqrt{A}}{\sqrt{0 - \ell}}}}} \]

      9. associate-/r/ N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{0 - V}}{\sqrt{A}} \cdot \sqrt{0 - \ell}}} \]

   6. Applied egg-rr 44.3%

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

   if -1e262 < (*.f64 V l) < -4.9999999999999996e-255

   1. Initial program 85.8%

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

      1. remove-double-div N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. inv-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. sqr-neg N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-pow N/A

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

      18. inv-pow N/A

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

      19. inv-pow N/A

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

      20. frac-2neg N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. clear-num N/A

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

      24. associate-/r* N/A

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

      25. neg-mul-1 N/A

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

      26. distribute-lft-neg-in N/A

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

   4. Applied egg-rr 78.6%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}} \]

      13. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}} \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      15. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      16. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      17. +-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\left(V \cdot \ell + 0\right)}}} \]

      18. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \left(\color{blue}{\ell \cdot V} + 0\right)}} \]

      19. accelerator-lowering-fma.f64 99.4

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\mathsf{fma}\left(\ell, V, 0\right)}}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}} \]

   if -4.9999999999999996e-255 < (*.f64 V l) < 0.0

   1. Initial program 35.9%

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

      1. remove-double-div N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. inv-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. sqr-neg N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-pow N/A

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

      18. inv-pow N/A

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

      19. inv-pow N/A

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

      20. frac-2neg N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. clear-num N/A

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

      24. associate-/r* N/A

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

      25. neg-mul-1 N/A

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

      26. distribute-lft-neg-in N/A

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

   4. Applied egg-rr 57.0%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. associate-*l/ N/A

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

      6. sqrt-div N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. div-inv N/A

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

      10. frac-2neg N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{\color{blue}{0 - \ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      14. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{\color{blue}{0 - \ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      16. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{0 - V}}} \]

      17. --lowering--.f64 41.1

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{0 - V}}} \]

   6. Applied egg-rr 41.1%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{0 - V}}} \]
   7. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{\mathsf{neg}\left(V\right)}}} \]

      2. neg-lowering-neg.f64 41.1

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{-V}}} \]

   8. Applied egg-rr 41.1%

      \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{0 - \ell}}}{\sqrt{\color{blue}{-V}}} \]

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+262}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-255}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - \frac{A}{\ell}}}{\sqrt{0 - V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-264}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (* (sqrt l) (sqrt (/ V A))))))
   (if (<= (* V l) -1e+262)
     t_0
     (if (<= (* V l) -5e-264)
       (* c0 (/ (sqrt (- 0.0 A)) (sqrt (- 0.0 (fma l V 0.0)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+289)
           (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
           (* c0 (sqrt (/ (/ A V) l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+262) {
		tmp = t_0;
	} else if ((V * l) <= -5e-264) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - fma(l, V, 0.0))));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+262)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-264)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(0.0 - fma(l, V, 0.0)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+262], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-264], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - N[(l * V + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e262 or -5.0000000000000001e-264 < (*.f64 V l) < 0.0

   1. Initial program 39.6%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 57.7

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

   4. Applied egg-rr 57.7%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\mathsf{neg}\left(V\right)}{\frac{A}{\color{blue}{0 - \ell}}}}} \]

      6. sqrt-undiv N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\mathsf{neg}\left(V\right)}}{\sqrt{\frac{A}{0 - \ell}}}}} \]

      7. sub0-neg N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{\color{blue}{0 - V}}}{\sqrt{\frac{A}{0 - \ell}}}} \]

      8. sqrt-undiv N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{0 - V}}{\color{blue}{\frac{\sqrt{A}}{\sqrt{0 - \ell}}}}} \]

      9. associate-/r/ N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{0 - V}}{\sqrt{A}} \cdot \sqrt{0 - \ell}}} \]

   6. Applied egg-rr 38.9%

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

   if -1e262 < (*.f64 V l) < -5.0000000000000001e-264

   1. Initial program 86.2%

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

      1. remove-double-div N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. inv-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. sqr-neg N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-pow N/A

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

      18. inv-pow N/A

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

      19. inv-pow N/A

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

      20. frac-2neg N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. clear-num N/A

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

      24. associate-/r* N/A

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

      25. neg-mul-1 N/A

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

      26. distribute-lft-neg-in N/A

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

   4. Applied egg-rr 79.2%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}} \]

      13. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}} \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V \cdot \ell\right)}}} \]

      15. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      16. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      17. +-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\left(V \cdot \ell + 0\right)}}} \]

      18. *-commutative N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \left(\color{blue}{\ell \cdot V} + 0\right)}} \]

      19. accelerator-lowering-fma.f64 99.4

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\mathsf{fma}\left(\ell, V, 0\right)}}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}} \]

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+262}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-264}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(\ell, V, 0\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 2e-313)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+280)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-313) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+280) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 2d-313) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+280) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-313) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+280) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 2e-313:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+280:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 2e-313)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+280)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 2e-313)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+280)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-313], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+280], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.99999999998e-313

   1. Initial program 39.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 55.4

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

   4. Applied egg-rr 55.4%

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

   if 1.99999999998e-313 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e280

   1. Initial program 99.2%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 90.6

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in V around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 99.3

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

   7. Simplified 99.3%

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

   if 2.0000000000000001e280 < (/.f64 A (*.f64 V l))

   1. Initial program 29.3%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 43.2

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

   4. Applied egg-rr 43.2%

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

Alternative 6: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+14)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) 0.0)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= (* V l) 1e+289)
       (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
       (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+14) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+14)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+14], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e14

   1. Initial program 70.3%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 47.8

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

   4. Applied egg-rr 47.8%

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

   if -1e14 < (*.f64 V l) < 0.0

   1. Initial program 61.2%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 67.6

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

   4. Applied egg-rr 67.6%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

Alternative 7: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) 1e+289)
     (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.3%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 68.4

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

   4. Applied egg-rr 68.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\mathsf{neg}\left(V\right)}{\frac{A}{\color{blue}{0 - \ell}}}}} \]

      6. sqrt-undiv N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\mathsf{neg}\left(V\right)}}{\sqrt{\frac{A}{0 - \ell}}}}} \]

      7. sub0-neg N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{\color{blue}{0 - V}}}{\sqrt{\frac{A}{0 - \ell}}}} \]

      8. sqrt-undiv N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{0 - V}}{\color{blue}{\frac{\sqrt{A}}{\sqrt{0 - \ell}}}}} \]

      9. associate-/r/ N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{0 - V}}{\sqrt{A}} \cdot \sqrt{0 - \ell}}} \]

   6. Applied egg-rr 44.1%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \left(\ell \cdot \frac{1}{A}\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (sqrt (* V (* l (/ 1.0 A)))))
   (if (<= (* V l) 1e+289)
     (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / sqrt((V * (l * (1.0 / A))));
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l * Float64(1.0 / A)))));
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.3%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 68.4

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

   4. Applied egg-rr 68.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 68.4

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

   6. Applied egg-rr 68.4%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \left(\ell \cdot \frac{1}{A}\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+289}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (/ c0 (sqrt (* V (/ l A))))
   (if (<= (* V l) 1e+289)
     (* c0 (/ (sqrt A) (sqrt (fma V l 0.0))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+289) {
		tmp = c0 * (sqrt(A) / sqrt(fma(V, l, 0.0)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+289)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(fma(V, l, 0.0))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+289], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.3%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 68.4

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

   4. Applied egg-rr 68.4%

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

   if 0.0 < (*.f64 V l) < 1.0000000000000001e289

   1. Initial program 80.4%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if 1.0000000000000001e289 < (*.f64 V l)

   1. Initial program 35.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 75.3

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

   4. Applied egg-rr 75.3%

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

Alternative 10: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-299}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-299)
   (/ c0 (sqrt (* V (/ l A))))
   (if (<= (* V l) 1e+287)
     (* (sqrt A) (/ c0 (sqrt (* V l))))
     (* c0 (sqrt (/ (* A (/ 1.0 V)) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-299) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+287) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A * (1.0 / V)) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 5d-299) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+287) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a * (1.0d0 / v)) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-299) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+287) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A * (1.0 / V)) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 5e-299:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+287:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A * (1.0 / V)) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-299)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+287)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A * Float64(1.0 / V)) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 5e-299)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+287)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A * (1.0 / V)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-299], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+287], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A * N[(1.0 / V), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 4.99999999999999956e-299

   1. Initial program 65.8%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 68.8

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

   4. Applied egg-rr 68.8%

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

   if 4.99999999999999956e-299 < (*.f64 V l) < 1.0000000000000001e287

   1. Initial program 79.7%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. +-lft-identity N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \cdot \sqrt{A} \]

      10. +-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \cdot \sqrt{A} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \cdot \sqrt{A} \]

      12. sqrt-lowering-sqrt.f64 96.8

          \[\leadsto \frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot \color{blue}{\sqrt{A}} \]

   4. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot \sqrt{A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 96.8

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

   6. Applied egg-rr 96.8%

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

   if 1.0000000000000001e287 < (*.f64 V l)

   1. Initial program 39.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 76.5

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

   4. Applied egg-rr 76.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 76.5

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

   6. Applied egg-rr 76.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-299}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-299}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-299)
   (/ c0 (sqrt (* V (/ l A))))
   (if (<= (* V l) 1e+287)
     (* (sqrt A) (/ c0 (sqrt (* V l))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-299) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+287) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= 5d-299) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+287) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-299) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+287) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 5e-299:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+287:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-299)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+287)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= 5e-299)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+287)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-299], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+287], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 4.99999999999999956e-299

   1. Initial program 65.8%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 68.8

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

   4. Applied egg-rr 68.8%

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

   if 4.99999999999999956e-299 < (*.f64 V l) < 1.0000000000000001e287

   1. Initial program 79.7%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. +-lft-identity N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \cdot \sqrt{A} \]

      10. +-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \cdot \sqrt{A} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \cdot \sqrt{A} \]

      12. sqrt-lowering-sqrt.f64 96.8

          \[\leadsto \frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot \color{blue}{\sqrt{A}} \]

   4. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot \sqrt{A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 96.8

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

   6. Applied egg-rr 96.8%

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

   if 1.0000000000000001e287 < (*.f64 V l)

   1. Initial program 39.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 76.5

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

   4. Applied egg-rr 76.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-299}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-209}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+33}:\\ \;\;\;\;c0 \cdot \frac{A}{\sqrt{A \cdot \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-209)
   (/ c0 (sqrt (* V (/ l A))))
   (if (<= (* V l) 4e+33)
     (* c0 (/ A (sqrt (* A (fma V l 0.0)))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-209) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 4e+33) {
		tmp = c0 * (A / sqrt((A * fma(V, l, 0.0))));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-209)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 4e+33)
		tmp = Float64(c0 * Float64(A / sqrt(Float64(A * fma(V, l, 0.0)))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-209], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+33], N[(c0 * N[(A / N[Sqrt[N[(A * N[(V * l + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 5.0000000000000005e-209

   1. Initial program 67.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   if 5.0000000000000005e-209 < (*.f64 V l) < 3.9999999999999998e33

   1. Initial program 69.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 54.2

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

   4. Applied egg-rr 54.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 54.2

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

   6. Applied egg-rr 54.2%

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

   8. Applied egg-rr 88.9%

      \[\leadsto c0 \cdot \color{blue}{\frac{A}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot A}}} \]

   if 3.9999999999999998e33 < (*.f64 V l)

   1. Initial program 71.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 79.0

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

   4. Applied egg-rr 79.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-209}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+33}:\\ \;\;\;\;c0 \cdot \frac{A}{\sqrt{A \cdot \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-209}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+33}:\\ \;\;\;\;c0 \cdot \frac{A}{\sqrt{A \cdot \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 5e-209)
   (* c0 (sqrt (/ (/ A l) V)))
   (if (<= (* V l) 4e+33)
     (* c0 (/ A (sqrt (* A (fma V l 0.0)))))
     (* c0 (sqrt (/ (/ A V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 5e-209) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if ((V * l) <= 4e+33) {
		tmp = c0 * (A / sqrt((A * fma(V, l, 0.0))));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 5e-209)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= 4e+33)
		tmp = Float64(c0 * Float64(A / sqrt(Float64(A * fma(V, l, 0.0)))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], 5e-209], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+33], N[(c0 * N[(A / N[Sqrt[N[(A * N[(V * l + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 5.0000000000000005e-209

   1. Initial program 67.5%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 69.5

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

   4. Applied egg-rr 69.5%

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

   if 5.0000000000000005e-209 < (*.f64 V l) < 3.9999999999999998e33

   1. Initial program 69.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 54.2

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

   4. Applied egg-rr 54.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 54.2

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

   6. Applied egg-rr 54.2%

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

   8. Applied egg-rr 88.9%

      \[\leadsto c0 \cdot \color{blue}{\frac{A}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot A}}} \]

   if 3.9999999999999998e33 < (*.f64 V l)

   1. Initial program 71.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 79.0

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

   4. Applied egg-rr 79.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 5 \cdot 10^{-209}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+33}:\\ \;\;\;\;c0 \cdot \frac{A}{\sqrt{A \cdot \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

Reproduce

herbie shell --seed 2024197 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))