Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.6% → 92.9%

Time: 11.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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-310)
   (* c0 (/ (sqrt (- 0.0 A)) (* (sqrt (- 0.0 V)) (sqrt l))))
   (if (<= (* V l) 4e-309)
     (/ c0 (sqrt (* l (/ V A))))
     (if (<= (* V l) 1e+296)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-310) {
		tmp = c0 * (sqrt((0.0 - A)) / (sqrt((0.0 - V)) * sqrt(l)));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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) <= (-1d-310)) then
        tmp = c0 * (sqrt((0.0d0 - a)) / (sqrt((0.0d0 - v)) * sqrt(l)))
    else if ((v * l) <= 4d-309) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 1d+296) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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) <= -1e-310) {
		tmp = c0 * (Math.sqrt((0.0 - A)) / (Math.sqrt((0.0 - V)) * Math.sqrt(l)));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-310:
		tmp = c0 * (math.sqrt((0.0 - A)) / (math.sqrt((0.0 - V)) * math.sqrt(l)))
	elif (V * l) <= 4e-309:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 1e+296:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	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-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / Float64(sqrt(Float64(0.0 - V)) * sqrt(l))));
	elseif (Float64(V * l) <= 4e-309)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 1e+296)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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) <= -1e-310)
		tmp = c0 * (sqrt((0.0 - A)) / (sqrt((0.0 - V)) * sqrt(l)));
	elseif ((V * l) <= 4e-309)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 1e+296)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	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], -1e-310], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-309], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+296], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.999999999999969e-311

   1. Initial program 77.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 74.4

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

   4. Applied egg-rr 74.4%

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

      1. un-div-inv N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. associate-/l/ N/A

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

      6. sqrt-prod N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. +-rgt-identity N/A

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

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

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

      14. neg-sub0 N/A

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

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

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 90.1

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

   6. Applied egg-rr 90.1%

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

      1. sub0-neg N/A

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

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

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

      5. sqrt-prod N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

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

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

      10. sqrt-lowering-sqrt.f64 48.1

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

   8. Applied egg-rr 48.1%

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

   if -9.999999999999969e-311 < (*.f64 V l) < 3.9999999999999977e-309

   1. Initial program 39.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.6

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

   4. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   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. associate-/r/ N/A

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

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

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

      5. /-lowering-/.f64 80.4

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

   6. Applied egg-rr 80.4%

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

   if 3.9999999999999977e-309 < (*.f64 V l) < 9.99999999999999981e295

   1. Initial program 86.9%

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

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

   5. Taylor expanded in V around 0

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

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

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

      2. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 9.99999999999999981e295 < (*.f64 V l)

   1. Initial program 29.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.8

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

   4. Applied egg-rr 72.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+304}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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+304)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -1e-304)
     (* c0 (/ (sqrt (- 0.0 A)) (sqrt (- 0.0 (* V l)))))
     (if (<= (* V l) 4e-309)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 1e+296)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+304) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -1e-304) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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+304)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= (-1d-304)) then
        tmp = c0 * (sqrt((0.0d0 - a)) / sqrt((0.0d0 - (v * l))))
    else if ((v * l) <= 4d-309) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 1d+296) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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+304) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= -1e-304) {
		tmp = c0 * (Math.sqrt((0.0 - A)) / Math.sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+304:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= -1e-304:
		tmp = c0 * (math.sqrt((0.0 - A)) / math.sqrt((0.0 - (V * l))))
	elif (V * l) <= 4e-309:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 1e+296:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	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+304)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -1e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(0.0 - Float64(V * l)))));
	elseif (Float64(V * l) <= 4e-309)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 1e+296)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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+304)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= -1e-304)
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - (V * l))));
	elseif ((V * l) <= 4e-309)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 1e+296)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	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+304], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-304], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-309], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+296], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999997e304

   1. Initial program 38.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 52.5

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

   4. Applied egg-rr 52.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. sqrt-prod N/A

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

      5. sqrt-undiv N/A

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

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

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

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

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

      8. sqrt-undiv N/A

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

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

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

      10. /-lowering-/.f64 49.8

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

   6. Applied egg-rr 49.8%

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

   if -4.9999999999999997e304 < (*.f64 V l) < -9.99999999999999971e-305

   1. Initial program 84.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 77.8

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

   4. Applied egg-rr 77.8%

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

      1. un-div-inv N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. associate-/l/ N/A

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

      6. sqrt-prod N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. +-rgt-identity N/A

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

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

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

      14. neg-sub0 N/A

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

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

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 99.5

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

   6. Applied egg-rr 99.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.5

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

   8. Applied egg-rr 99.5%

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

   if -9.99999999999999971e-305 < (*.f64 V l) < 3.9999999999999977e-309

   1. Initial program 41.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 81.1

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

   4. Applied egg-rr 81.1%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   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. associate-/r/ N/A

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

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

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

      5. /-lowering-/.f64 80.9

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

   6. Applied egg-rr 80.9%

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

   if 3.9999999999999977e-309 < (*.f64 V l) < 9.99999999999999981e295

   1. Initial program 86.9%

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

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

   5. Taylor expanded in V around 0

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

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

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

      2. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 9.99999999999999981e295 < (*.f64 V l)

   1. Initial program 29.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.8

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

   4. Applied egg-rr 72.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+304}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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+304)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-304)
     (* c0 (/ (sqrt (- 0.0 A)) (sqrt (- 0.0 (* V l)))))
     (if (<= (* V l) 4e-309)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 1e+296)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+304) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-304) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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+304)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-1d-304)) then
        tmp = c0 * (sqrt((0.0d0 - a)) / sqrt((0.0d0 - (v * l))))
    else if ((v * l) <= 4d-309) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 1d+296) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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+304) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-304) {
		tmp = c0 * (Math.sqrt((0.0 - A)) / Math.sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+304:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-304:
		tmp = c0 * (math.sqrt((0.0 - A)) / math.sqrt((0.0 - (V * l))))
	elif (V * l) <= 4e-309:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 1e+296:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	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+304)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(0.0 - Float64(V * l)))));
	elseif (Float64(V * l) <= 4e-309)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 1e+296)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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+304)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-304)
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - (V * l))));
	elseif ((V * l) <= 4e-309)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 1e+296)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	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+304], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-304], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-309], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+296], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.9999999999999997e304

   1. Initial program 38.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 49.9

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

   4. Applied egg-rr 49.9%

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

   if -4.9999999999999997e304 < (*.f64 V l) < -9.99999999999999971e-305

   1. Initial program 84.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 77.8

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

   4. Applied egg-rr 77.8%

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

      1. un-div-inv N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. associate-/l/ N/A

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

      6. sqrt-prod N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. +-rgt-identity N/A

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

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

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

      14. neg-sub0 N/A

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

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

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 99.5

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

   6. Applied egg-rr 99.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.5

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

   8. Applied egg-rr 99.5%

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

   if -9.99999999999999971e-305 < (*.f64 V l) < 3.9999999999999977e-309

   1. Initial program 41.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 81.1

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

   4. Applied egg-rr 81.1%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   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. associate-/r/ N/A

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

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

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

      5. /-lowering-/.f64 80.9

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

   6. Applied egg-rr 80.9%

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

   if 3.9999999999999977e-309 < (*.f64 V l) < 9.99999999999999981e295

   1. Initial program 86.9%

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

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

   5. Taylor expanded in V around 0

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

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

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

      2. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 9.99999999999999981e295 < (*.f64 V l)

   1. Initial program 29.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.8

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

   4. Applied egg-rr 72.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(\ell, V, 0\right)}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 5e-313)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+275)
       (/ c0 (sqrt (/ (fma l V 0.0) 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 <= 5e-313) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+275) {
		tmp = c0 / sqrt((fma(l, V, 0.0) / A));
	} else {
		tmp = c0 / 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 <= 5e-313)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+275)
		tmp = Float64(c0 / sqrt(Float64(fma(l, V, 0.0) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-313], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+275], N[(c0 / N[Sqrt[N[(N[(l * V + 0.0), $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)) < 5.00000000002e-313

   1. Initial program 32.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 51.6

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

   4. Applied egg-rr 51.6%

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

   if 5.00000000002e-313 < (/.f64 A (*.f64 V l)) < 1.99999999999999992e275

   1. Initial program 99.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 89.1

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

   4. Applied egg-rr 89.1%

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

      1. un-div-inv N/A

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

      2. sqrt-undiv N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. sqrt-div N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l/ N/A

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

      8. sqrt-prod N/A

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

      9. +-rgt-identity N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sqrt-undiv N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 99.3

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

   6. Applied egg-rr 99.3%

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

   if 1.99999999999999992e275 < (/.f64 A (*.f64 V l))

   1. Initial program 45.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 65.9

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

   4. Applied egg-rr 65.9%

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

Alternative 5: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+271) (* c0 (sqrt t_0)) (/ 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 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+271) {
		tmp = c0 * sqrt(t_0);
	} 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 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+271) then
        tmp = c0 * sqrt(t_0)
    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 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+271) {
		tmp = c0 * Math.sqrt(t_0);
	} 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 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+271:
		tmp = c0 * math.sqrt(t_0)
	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 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+271)
		tmp = Float64(c0 * sqrt(t_0));
	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 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+271)
		tmp = c0 * sqrt(t_0);
	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, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+271], N[(c0 * N[Sqrt[t$95$0], $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)) < 0.0

   1. Initial program 31.2%

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

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999991e271

   1. Initial program 98.7%

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

   if 1.99999999999999991e271 < (/.f64 A (*.f64 V l))

   1. Initial program 46.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 66.4

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

   4. Applied egg-rr 66.4%

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

Alternative 6: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ A (* V l))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 1e+296) (* c0 (sqrt 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 = 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+296) {
		tmp = c0 * sqrt(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 = a / (v * l)
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+296) then
        tmp = c0 * sqrt(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 = 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+296) {
		tmp = c0 * Math.sqrt(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 = 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+296:
		tmp = c0 * math.sqrt(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(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+296)
		tmp = Float64(c0 * sqrt(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 = 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+296)
		tmp = c0 * sqrt(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[(A / N[(V * l), $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+296], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 9.99999999999999981e295 < (/.f64 A (*.f64 V l))

   1. Initial program 33.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 57.2

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

   4. Applied egg-rr 57.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.99999999999999981e295

   1. Initial program 98.7%

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

Alternative 7: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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-304)
   (* c0 (/ (sqrt (- 0.0 A)) (sqrt (- 0.0 (* V l)))))
   (if (<= (* V l) 4e-309)
     (/ c0 (sqrt (* l (/ V A))))
     (if (<= (* V l) 1e+296)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-304) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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) <= (-1d-304)) then
        tmp = c0 * (sqrt((0.0d0 - a)) / sqrt((0.0d0 - (v * l))))
    else if ((v * l) <= 4d-309) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 1d+296) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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) <= -1e-304) {
		tmp = c0 * (Math.sqrt((0.0 - A)) / Math.sqrt((0.0 - (V * l))));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-304:
		tmp = c0 * (math.sqrt((0.0 - A)) / math.sqrt((0.0 - (V * l))))
	elif (V * l) <= 4e-309:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 1e+296:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	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-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(0.0 - Float64(V * l)))));
	elseif (Float64(V * l) <= 4e-309)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 1e+296)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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) <= -1e-304)
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - (V * l))));
	elseif ((V * l) <= 4e-309)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 1e+296)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	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], -1e-304], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-309], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+296], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999971e-305

   1. Initial program 77.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 74.1

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

   4. Applied egg-rr 74.1%

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

      1. un-div-inv N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. associate-/l/ N/A

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

      6. sqrt-prod N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. +-rgt-identity N/A

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

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

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

      14. neg-sub0 N/A

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

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

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 90.1

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

   6. Applied egg-rr 90.1%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 90.1

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

   8. Applied egg-rr 90.1%

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

   if -9.99999999999999971e-305 < (*.f64 V l) < 3.9999999999999977e-309

   1. Initial program 41.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 81.1

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

   4. Applied egg-rr 81.1%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   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. associate-/r/ N/A

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

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

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

      5. /-lowering-/.f64 80.9

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

   6. Applied egg-rr 80.9%

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

   if 3.9999999999999977e-309 < (*.f64 V l) < 9.99999999999999981e295

   1. Initial program 86.9%

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

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

   5. Taylor expanded in V around 0

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

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

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

      2. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 9.99999999999999981e295 < (*.f64 V l)

   1. Initial program 29.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.8

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

   4. Applied egg-rr 72.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+64}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

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) -2e+64)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* V l) 4e-309)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= (* V l) 1e+296)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+64) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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) <= (-2d+64)) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 4d-309) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+296) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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) <= -2e+64) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 4e-309) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+296) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+64:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 4e-309:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+296:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	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) <= -2e+64)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 4e-309)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+296)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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) <= -2e+64)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 4e-309)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+296)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / l) / V));
	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], -2e+64], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-309], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+296], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000004e64

   1. Initial program 59.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 60.4

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

   4. Applied egg-rr 60.4%

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

   if -2.00000000000000004e64 < (*.f64 V l) < 3.9999999999999977e-309

   1. Initial program 70.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.2

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

   4. Applied egg-rr 80.2%

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

   if 3.9999999999999977e-309 < (*.f64 V l) < 9.99999999999999981e295

   1. Initial program 86.9%

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

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

   5. Taylor expanded in V around 0

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

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

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

      2. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   if 9.99999999999999981e295 < (*.f64 V l)

   1. Initial program 29.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.8

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

   4. Applied egg-rr 72.8%

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

Alternative 9: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - \frac{A}{\ell}}}{\sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \end{array} \]

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 -2e-310)
   (* c0 (/ (sqrt (- 0.0 (/ A l))) (sqrt (- 0.0 V))))
   (* c0 (/ (sqrt A) (* (sqrt l) (sqrt V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2e-310) {
		tmp = c0 * (sqrt((0.0 - (A / l))) / sqrt((0.0 - V)));
	} else {
		tmp = c0 * (sqrt(A) / (sqrt(l) * sqrt(V)));
	}
	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 <= (-2d-310)) then
        tmp = c0 * (sqrt((0.0d0 - (a / l))) / sqrt((0.0d0 - v)))
    else
        tmp = c0 * (sqrt(a) / (sqrt(l) * sqrt(v)))
    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 <= -2e-310) {
		tmp = c0 * (Math.sqrt((0.0 - (A / l))) / Math.sqrt((0.0 - V)));
	} else {
		tmp = c0 * (Math.sqrt(A) / (Math.sqrt(l) * Math.sqrt(V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -2e-310:
		tmp = c0 * (math.sqrt((0.0 - (A / l))) / math.sqrt((0.0 - V)))
	else:
		tmp = c0 * (math.sqrt(A) / (math.sqrt(l) * math.sqrt(V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -2e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - Float64(A / l))) / sqrt(Float64(0.0 - V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / Float64(sqrt(l) * sqrt(V))));
	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 <= -2e-310)
		tmp = c0 * (sqrt((0.0 - (A / l))) / sqrt((0.0 - V)));
	else
		tmp = c0 * (sqrt(A) / (sqrt(l) * sqrt(V)));
	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[V, -2e-310], N[(c0 * N[(N[Sqrt[N[(0.0 - N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -1.999999999999994e-310

   1. Initial program 74.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 76.0

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

   4. Applied egg-rr 76.0%

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

      1. un-div-inv N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. sqrt-div N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

      13. neg-sub0 N/A

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

      14. --lowering--.f64 85.0

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

   6. Applied egg-rr 85.0%

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

   if -1.999999999999994e-310 < V

   1. Initial program 70.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 43.2

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

   4. Applied egg-rr 43.2%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 56.2

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

   6. Applied egg-rr 56.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - \frac{A}{\ell}}}{\sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.6% 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 72.3%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))